Properties

Label 12-2738e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.213\times 10^{20}$
Sign $1$
Analytic cond. $1.09210\times 10^{8}$
Root an. cond. $4.67579$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 21·4-s + 6·5-s − 56·8-s − 12·9-s − 36·10-s − 6·11-s + 12·13-s + 126·16-s + 12·17-s + 72·18-s + 12·19-s + 126·20-s + 36·22-s − 6·23-s + 6·25-s − 72·26-s − 2·27-s − 6·29-s + 18·31-s − 252·32-s − 72·34-s − 252·36-s − 72·38-s − 336·40-s − 12·41-s − 126·44-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s + 2.68·5-s − 19.7·8-s − 4·9-s − 11.3·10-s − 1.80·11-s + 3.32·13-s + 63/2·16-s + 2.91·17-s + 16.9·18-s + 2.75·19-s + 28.1·20-s + 7.67·22-s − 1.25·23-s + 6/5·25-s − 14.1·26-s − 0.384·27-s − 1.11·29-s + 3.23·31-s − 44.5·32-s − 12.3·34-s − 42·36-s − 11.6·38-s − 53.1·40-s − 1.87·41-s − 18.9·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(1.09210\times 10^{8}\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 37^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.233494327\)
\(L(\frac12)\) \(\approx\) \(1.233494327\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
37 \( 1 \)
good3 \( ( 1 + 2 p T^{2} + T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} )^{2} \)
5 \( 1 - 6 T + 6 p T^{2} - 108 T^{3} + 66 p T^{4} - 174 p T^{5} + 2059 T^{6} - 174 p^{2} T^{7} + 66 p^{3} T^{8} - 108 p^{3} T^{9} + 6 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 27 T^{2} - 2 T^{3} + 351 T^{4} - 54 T^{5} + 2955 T^{6} - 54 p T^{7} + 351 p^{2} T^{8} - 2 p^{3} T^{9} + 27 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 + 6 T + 39 T^{2} + 180 T^{3} + 75 p T^{4} + 2958 T^{5} + 10735 T^{6} + 2958 p T^{7} + 75 p^{3} T^{8} + 180 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 12 T + 9 p T^{2} - 750 T^{3} + 4275 T^{4} - 19020 T^{5} + 76535 T^{6} - 19020 p T^{7} + 4275 p^{2} T^{8} - 750 p^{3} T^{9} + 9 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 12 T + 129 T^{2} - 900 T^{3} + 5694 T^{4} - 28236 T^{5} + 128869 T^{6} - 28236 p T^{7} + 5694 p^{2} T^{8} - 900 p^{3} T^{9} + 129 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 12 T + 123 T^{2} - 696 T^{3} + 195 p T^{4} - 13260 T^{5} + 62471 T^{6} - 13260 p T^{7} + 195 p^{3} T^{8} - 696 p^{3} T^{9} + 123 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 93 T^{2} + 324 T^{3} + 3237 T^{4} + 6864 T^{5} + 75499 T^{6} + 6864 p T^{7} + 3237 p^{2} T^{8} + 324 p^{3} T^{9} + 93 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 156 T^{2} + 756 T^{3} + 10590 T^{4} + 41028 T^{5} + 400939 T^{6} + 41028 p T^{7} + 10590 p^{2} T^{8} + 756 p^{3} T^{9} + 156 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 18 T + 243 T^{2} - 2250 T^{3} + 18603 T^{4} - 124740 T^{5} + 763739 T^{6} - 124740 p T^{7} + 18603 p^{2} T^{8} - 2250 p^{3} T^{9} + 243 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 210 T^{2} + 1692 T^{3} + 16863 T^{4} + 105528 T^{5} + 815164 T^{6} + 105528 p T^{7} + 16863 p^{2} T^{8} + 1692 p^{3} T^{9} + 210 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 180 T^{2} - 144 T^{3} + 15732 T^{4} - 14220 T^{5} + 847835 T^{6} - 14220 p T^{7} + 15732 p^{2} T^{8} - 144 p^{3} T^{9} + 180 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 6 T + 165 T^{2} + 810 T^{3} + 15090 T^{4} + 1290 p T^{5} + 852109 T^{6} + 1290 p^{2} T^{7} + 15090 p^{2} T^{8} + 810 p^{3} T^{9} + 165 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 327 T^{2} + 2538 T^{3} + 12975 T^{4} + 28122 T^{5} + 48247 T^{6} + 28122 p T^{7} + 12975 p^{2} T^{8} + 2538 p^{3} T^{9} + 327 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 219 T^{2} - 1224 T^{3} + 255 p T^{4} - 44292 T^{5} + 793591 T^{6} - 44292 p T^{7} + 255 p^{3} T^{8} - 1224 p^{3} T^{9} + 219 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 12 T + 294 T^{2} - 2316 T^{3} + 34071 T^{4} - 3288 p T^{5} + 2423252 T^{6} - 3288 p^{2} T^{7} + 34071 p^{2} T^{8} - 2316 p^{3} T^{9} + 294 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 279 T^{2} - 3364 T^{3} + 40653 T^{4} - 375282 T^{5} + 3299835 T^{6} - 375282 p T^{7} + 40653 p^{2} T^{8} - 3364 p^{3} T^{9} + 279 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 24 T + 282 T^{2} - 1944 T^{3} + 15855 T^{4} - 199680 T^{5} + 2156812 T^{6} - 199680 p T^{7} + 15855 p^{2} T^{8} - 1944 p^{3} T^{9} + 282 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 72 T^{2} - 322 T^{3} + 1368 T^{4} + 44892 T^{5} + 304995 T^{6} + 44892 p T^{7} + 1368 p^{2} T^{8} - 322 p^{3} T^{9} + 72 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 12 T + 513 T^{2} - 4728 T^{3} + 105678 T^{4} - 745644 T^{5} + 11279717 T^{6} - 745644 p T^{7} + 105678 p^{2} T^{8} - 4728 p^{3} T^{9} + 513 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T + 183 T^{2} + 1080 T^{3} + 14937 T^{4} + 92382 T^{5} + 1102075 T^{6} + 92382 p T^{7} + 14937 p^{2} T^{8} + 1080 p^{3} T^{9} + 183 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 291 T^{2} - 972 T^{3} + 40137 T^{4} - 223884 T^{5} + 3970519 T^{6} - 223884 p T^{7} + 40137 p^{2} T^{8} - 972 p^{3} T^{9} + 291 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 12 T + 342 T^{2} + 2892 T^{3} + 50175 T^{4} + 364584 T^{5} + 5289236 T^{6} + 364584 p T^{7} + 50175 p^{2} T^{8} + 2892 p^{3} T^{9} + 342 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.03833785967616919201085991498, −4.33603308451560287074535349640, −4.32712548064284588072529358773, −3.80996062064349031000753653424, −3.59791727596950176744909823421, −3.52493988950045521373096219506, −3.48425798936028994544415541333, −3.25996625770374851074909796686, −3.17072185159364999825269323582, −3.13040105567542438981221850459, −3.02531474541251843794735808554, −2.72497252615692103542402380786, −2.71459229533343755092926362443, −2.27837501298876929263722725891, −2.14548418551412680734967545585, −2.04993561278822157216565943574, −1.92857598976197342247850030590, −1.76828390970632740854588967999, −1.64536106920158994875112925863, −1.32599805876772838240625982384, −1.10871814044735780767221254251, −0.925480169397918778280261699795, −0.70567350604814230070825810532, −0.41072755429299498553268754878, −0.37322233515878357090259918000, 0.37322233515878357090259918000, 0.41072755429299498553268754878, 0.70567350604814230070825810532, 0.925480169397918778280261699795, 1.10871814044735780767221254251, 1.32599805876772838240625982384, 1.64536106920158994875112925863, 1.76828390970632740854588967999, 1.92857598976197342247850030590, 2.04993561278822157216565943574, 2.14548418551412680734967545585, 2.27837501298876929263722725891, 2.71459229533343755092926362443, 2.72497252615692103542402380786, 3.02531474541251843794735808554, 3.13040105567542438981221850459, 3.17072185159364999825269323582, 3.25996625770374851074909796686, 3.48425798936028994544415541333, 3.52493988950045521373096219506, 3.59791727596950176744909823421, 3.80996062064349031000753653424, 4.32712548064284588072529358773, 4.33603308451560287074535349640, 5.03833785967616919201085991498

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.