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Properties

Label 2-624-13.12-c5-0-61
Degree 22
Conductor 624624
Sign −0.373+0.927i-0.373 + 0.927i
Analytic cond. 100.079100.079
Root an. cond. 10.003910.0039
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 9.73i·5-s − 105. i·7-s + 81·9-s − 269. i·11-s + (227. − 565. i)13-s − 87.6i·15-s + 1.66e3·17-s + 2.82i·19-s − 946. i·21-s − 2.15e3·23-s + 3.03e3·25-s + 729·27-s + 220.·29-s + 788. i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.174i·5-s − 0.811i·7-s + 0.333·9-s − 0.671i·11-s + (0.373 − 0.927i)13-s − 0.100i·15-s + 1.39·17-s + 0.00179i·19-s − 0.468i·21-s − 0.847·23-s + 0.969·25-s + 0.192·27-s + 0.0487·29-s + 0.147i·31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.373+0.927i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=((−0.373+0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.373+0.927i-0.373 + 0.927i
Analytic conductor: 100.079100.079
Root analytic conductor: 10.003910.0039
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ624(337,⋅)\chi_{624} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :5/2), −0.373+0.927i)(2,\ 624,\ (\ :5/2),\ -0.373 + 0.927i)

Particular Values

L(3)L(3) ≈\approx 2.5008133922.500813392
L(12)L(\frac12) ≈\approx 2.5008133922.500813392
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−9T 1 - 9T
13 1+(−227.+565.i)T 1 + (-227. + 565. i)T
good5 1+9.73iT−3.12e3T2 1 + 9.73iT - 3.12e3T^{2}
7 1+105.iT−1.68e4T2 1 + 105. iT - 1.68e4T^{2}
11 1+269.iT−1.61e5T2 1 + 269. iT - 1.61e5T^{2}
17 1−1.66e3T+1.41e6T2 1 - 1.66e3T + 1.41e6T^{2}
19 1−2.82iT−2.47e6T2 1 - 2.82iT - 2.47e6T^{2}
23 1+2.15e3T+6.43e6T2 1 + 2.15e3T + 6.43e6T^{2}
29 1−220.T+2.05e7T2 1 - 220.T + 2.05e7T^{2}
31 1−788.iT−2.86e7T2 1 - 788. iT - 2.86e7T^{2}
37 1+980.iT−6.93e7T2 1 + 980. iT - 6.93e7T^{2}
41 1−1.48e4iT−1.15e8T2 1 - 1.48e4iT - 1.15e8T^{2}
43 1+1.41e4T+1.47e8T2 1 + 1.41e4T + 1.47e8T^{2}
47 1+2.51e4iT−2.29e8T2 1 + 2.51e4iT - 2.29e8T^{2}
53 1+3.09e4T+4.18e8T2 1 + 3.09e4T + 4.18e8T^{2}
59 1+2.50e4iT−7.14e8T2 1 + 2.50e4iT - 7.14e8T^{2}
61 1−1.20e4T+8.44e8T2 1 - 1.20e4T + 8.44e8T^{2}
67 1+1.44e4iT−1.35e9T2 1 + 1.44e4iT - 1.35e9T^{2}
71 1−3.35e4iT−1.80e9T2 1 - 3.35e4iT - 1.80e9T^{2}
73 1+2.58e4iT−2.07e9T2 1 + 2.58e4iT - 2.07e9T^{2}
79 1+1.56e4T+3.07e9T2 1 + 1.56e4T + 3.07e9T^{2}
83 1+2.81e3iT−3.93e9T2 1 + 2.81e3iT - 3.93e9T^{2}
89 1+1.62e4iT−5.58e9T2 1 + 1.62e4iT - 5.58e9T^{2}
97 1+1.26e5iT−8.58e9T2 1 + 1.26e5iT - 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.702959958826889378390510826417, −8.429783374204288222139686105920, −8.027760978500492930722920302208, −7.02588150096757444305266171911, −5.93645725721309801042552248668, −4.91592431606194350669327606818, −3.66743076767519357441118507892, −3.05000605660656356164343634279, −1.45750210164007715785495144194, −0.50701195193075504634761714042, 1.35578923093332468621247579187, 2.35851593384745927861754297867, 3.40371240542544688178072234317, 4.48709561893140109337709841220, 5.59506836407597667101801719221, 6.59672842417611076446066574968, 7.54593360723733782187989035426, 8.413533170859670448526809533543, 9.249422296702242575168032316724, 9.913490408961604823672302631685

Graph of the ZZ-function along the critical line