Properties

Label 24-1323e12-1.1-c1e12-0-0
Degree $24$
Conductor $2.876\times 10^{37}$
Sign $1$
Analytic cond. $1.93216\times 10^{12}$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 2·16-s − 6·25-s − 16·37-s + 20·43-s − 6·64-s − 72·67-s + 72·79-s − 24·100-s − 40·109-s + 58·121-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 80·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·4-s + 1/2·16-s − 6/5·25-s − 2.63·37-s + 3.04·43-s − 3/4·64-s − 8.79·67-s + 8.10·79-s − 2.39·100-s − 3.83·109-s + 5.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 6.09·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{36} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(1.93216\times 10^{12}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{36} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9961433338\)
\(L(\frac12)\) \(\approx\) \(0.9961433338\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - p T^{2} + 5 T^{4} - 11 T^{6} + 5 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \)
5 \( ( 1 + 3 T^{2} + 33 T^{4} + 187 T^{6} + 33 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 29 T^{2} + 401 T^{4} - 4241 T^{6} + 401 p^{2} T^{8} - 29 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 36 T^{2} + 720 T^{4} - 10271 T^{6} + 720 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 72 T^{2} + 2520 T^{4} + 53647 T^{6} + 2520 p^{2} T^{8} + 72 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 - 63 T^{2} + 2169 T^{4} - 50411 T^{6} + 2169 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 44 T^{2} - 196 T^{4} + 31963 T^{6} - 196 p^{2} T^{8} - 44 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 137 T^{2} + 8645 T^{4} - 319673 T^{6} + 8645 p^{2} T^{8} - 137 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 48 T^{2} + 3036 T^{4} - 90731 T^{6} + 3036 p^{2} T^{8} - 48 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 4 T + 92 T^{2} + 229 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
41 \( ( 1 + 132 T^{2} + 8472 T^{4} + 388519 T^{6} + 8472 p^{2} T^{8} + 132 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 - 5 T + 113 T^{2} - 431 T^{3} + 113 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
47 \( ( 1 + 147 T^{2} + 11625 T^{4} + 631267 T^{6} + 11625 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 200 T^{2} + 18836 T^{4} - 1170761 T^{6} + 18836 p^{2} T^{8} - 200 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 + 291 T^{2} + 38085 T^{4} + 2876659 T^{6} + 38085 p^{2} T^{8} + 291 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 237 T^{2} + 29262 T^{4} - 2210249 T^{6} + 29262 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 + 18 T + 216 T^{2} + 1735 T^{3} + 216 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
71 \( ( 1 - 341 T^{2} + 53765 T^{4} - 4892609 T^{6} + 53765 p^{2} T^{8} - 341 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 243 T^{2} + 33777 T^{4} - 2942435 T^{6} + 33777 p^{2} T^{8} - 243 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 18 T + 252 T^{2} - 2837 T^{3} + 252 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 + 339 T^{2} + 56361 T^{4} + 5822683 T^{6} + 56361 p^{2} T^{8} + 339 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 333 T^{2} + 56934 T^{4} + 6181081 T^{6} + 56934 p^{2} T^{8} + 333 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 378 T^{2} + 65727 T^{4} - 7461452 T^{6} + 65727 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.01371979984636755571761968739, −2.87855309289083544190784370148, −2.80047228523413379968931986926, −2.66258477719944300686322350989, −2.65030383181059154347683321725, −2.61452226453338698656337230744, −2.39140283669171697157005521038, −2.32605997206898775737152561212, −2.22312869077430032336123706928, −2.20951611826878483646300072927, −2.15610915989791104181754962439, −2.07158984630470129149480897224, −1.74277712529403555390153935360, −1.70855010363566302671780848008, −1.68291458774730828601538410132, −1.51972642147644876863059866097, −1.48178921500680867280685967511, −1.32996791150708728023108016905, −1.29686330435230562645981325753, −0.980921288260781301466566197420, −0.934795940253944109852513375695, −0.55936531763959491094071632986, −0.53060830915269153037286455638, −0.39526755961886483477677977041, −0.05721754742115903492735777805, 0.05721754742115903492735777805, 0.39526755961886483477677977041, 0.53060830915269153037286455638, 0.55936531763959491094071632986, 0.934795940253944109852513375695, 0.980921288260781301466566197420, 1.29686330435230562645981325753, 1.32996791150708728023108016905, 1.48178921500680867280685967511, 1.51972642147644876863059866097, 1.68291458774730828601538410132, 1.70855010363566302671780848008, 1.74277712529403555390153935360, 2.07158984630470129149480897224, 2.15610915989791104181754962439, 2.20951611826878483646300072927, 2.22312869077430032336123706928, 2.32605997206898775737152561212, 2.39140283669171697157005521038, 2.61452226453338698656337230744, 2.65030383181059154347683321725, 2.66258477719944300686322350989, 2.80047228523413379968931986926, 2.87855309289083544190784370148, 3.01371979984636755571761968739

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

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