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Properties

Label 24-56e12-1.1-c1e12-0-0
Degree $24$
Conductor $9.512\times 10^{20}$
Sign $1$
Analytic cond. $6.39118\times 10^{-5}$
Root an. cond. $0.668701$
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  − 6·3-s − 4·8-s + 9·9-s − 6·11-s − 6·17-s − 6·19-s + 24·24-s + 15·25-s + 18·27-s + 36·33-s − 6·49-s + 36·51-s + 36·57-s + 42·59-s − 4·64-s + 30·67-s − 36·72-s + 18·73-s − 90·75-s − 66·81-s + 24·88-s + 18·89-s − 54·99-s − 18·107-s + 24·113-s + 69·121-s + 127-s + ⋯
L(s)  = 1  − 3.46·3-s − 1.41·8-s + 3·9-s − 1.80·11-s − 1.45·17-s − 1.37·19-s + 4.89·24-s + 3·25-s + 3.46·27-s + 6.26·33-s − 6/7·49-s + 5.04·51-s + 4.76·57-s + 5.46·59-s − 1/2·64-s + 3.66·67-s − 4.24·72-s + 2.10·73-s − 10.3·75-s − 7.33·81-s + 2.55·88-s + 1.90·89-s − 5.42·99-s − 1.74·107-s + 2.25·113-s + 6.27·121-s + 0.0887·127-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{36} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(6.39118\times 10^{-5}\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{36} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.03247839470\)
\(L(\frac12)\) \(\approx\) \(0.03247839470\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T^{3} + p^{3} T^{6} )^{2} \)
7 \( 1 + 6 T^{2} - 33 T^{4} - 100 p T^{6} - 33 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \)
good3 \( ( 1 + p T + p^{2} T^{2} + 2 p^{2} T^{3} + 11 p T^{4} + 7 p^{2} T^{5} + 34 p T^{6} + 7 p^{3} T^{7} + 11 p^{3} T^{8} + 2 p^{5} T^{9} + p^{6} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \)
5 \( 1 - 3 p T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 7989 p^{2} T^{14} + 1641 p^{4} T^{16} - 412 p^{6} T^{18} + 99 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \)
11 \( ( 1 + 3 T - 21 T^{2} - 28 T^{3} + 393 T^{4} + 153 T^{5} - 4846 T^{6} + 153 p T^{7} + 393 p^{2} T^{8} - 28 p^{3} T^{9} - 21 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
13 \( ( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 903 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 729 T^{4} + 753 T^{5} + 12014 T^{6} + 753 p T^{7} + 729 p^{2} T^{8} + 108 p^{3} T^{9} + 39 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 705 T^{4} + 2265 T^{5} + 12706 T^{6} + 2265 p T^{7} + 705 p^{2} T^{8} + 108 p^{3} T^{9} + 39 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 3 p T^{2} + 81 p T^{4} + 44136 T^{6} + 64755 p T^{8} + 1458309 p T^{10} + 591428630 T^{12} + 1458309 p^{3} T^{14} + 64755 p^{5} T^{16} + 44136 p^{6} T^{18} + 81 p^{9} T^{20} + 3 p^{11} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 7623 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( 1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 87513459 p^{2} T^{14} - 455595 p^{4} T^{16} - 78332 p^{6} T^{18} + 3867 p^{8} T^{20} - 87 p^{10} T^{22} + p^{12} T^{24} \)
37 \( 1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 501483003 p^{2} T^{14} + 14203497 p^{4} T^{16} + 152764 p^{6} T^{18} + 3339 p^{8} T^{20} + 105 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 6783 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + p T^{2} )^{12} \)
47 \( 1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 74583429 p^{2} T^{14} + 18269013 p^{4} T^{16} - 642364 p^{6} T^{18} + 13131 p^{8} T^{20} - 159 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 1185419067 p^{2} T^{14} + 39259017 p^{4} T^{16} - 10308 p^{6} T^{18} - 693 p^{8} T^{20} + 105 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 - 21 T + 309 T^{2} - 3402 T^{3} + 29601 T^{4} - 225789 T^{5} + 1760606 T^{6} - 225789 p T^{7} + 29601 p^{2} T^{8} - 3402 p^{3} T^{9} + 309 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( 1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 3286587357 p^{2} T^{14} + 54786249 p^{4} T^{16} - 352004 p^{6} T^{18} + 7419 p^{8} T^{20} - 159 p^{10} T^{22} + p^{12} T^{24} \)
67 \( ( 1 - 15 T - 3 T^{2} + 142 T^{3} + 9993 T^{4} - 15123 T^{5} - 719466 T^{6} - 15123 p T^{7} + 9993 p^{2} T^{8} + 142 p^{3} T^{9} - 3 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 16 T + p T^{2} )^{6}( 1 + 16 T + p T^{2} )^{6} \)
73 \( ( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 147195 p T^{7} + 16701 p^{2} T^{8} - 1404 p^{3} T^{9} + 183 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( 1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 18988265187 p^{2} T^{14} + 267713397 p^{4} T^{16} + 2829272 p^{6} T^{18} + 30519 p^{8} T^{20} + 261 p^{10} T^{22} + p^{12} T^{24} \)
83 \( ( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 28455 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{6} \)
97 \( ( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 20703 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−5.88071359817964704063112862272, −5.71214577969151355896233847414, −5.58242304557066225359821779201, −5.57637555101040971646264820839, −5.33006556147870570644606568628, −5.28750137418608858390678233893, −5.17630353464326626934445769754, −5.05122156712507315950861322546, −5.02882862742591161467975464875, −4.78194298745032568164569571545, −4.62309434804969609022274494241, −4.49840710114830048460179729787, −4.31549444666244711267873393724, −4.12153505613780386397733972440, −3.71851100473250361583582482992, −3.64746024676194995257560961571, −3.44036712928723524042188301584, −3.42421322678768620288794940431, −3.06280200854989224234076221949, −2.81034260296741703363706522537, −2.52718474427026652349664480756, −2.38061587884815147924623858364, −2.29130353705919315743837983486, −2.15983832251087429689402678176, −0.76605765556851903447854019152, 0.76605765556851903447854019152, 2.15983832251087429689402678176, 2.29130353705919315743837983486, 2.38061587884815147924623858364, 2.52718474427026652349664480756, 2.81034260296741703363706522537, 3.06280200854989224234076221949, 3.42421322678768620288794940431, 3.44036712928723524042188301584, 3.64746024676194995257560961571, 3.71851100473250361583582482992, 4.12153505613780386397733972440, 4.31549444666244711267873393724, 4.49840710114830048460179729787, 4.62309434804969609022274494241, 4.78194298745032568164569571545, 5.02882862742591161467975464875, 5.05122156712507315950861322546, 5.17630353464326626934445769754, 5.28750137418608858390678233893, 5.33006556147870570644606568628, 5.57637555101040971646264820839, 5.58242304557066225359821779201, 5.71214577969151355896233847414, 5.88071359817964704063112862272

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

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