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Properties

Label 2-624-13.12-c5-0-26
Degree 22
Conductor 624624
Sign −0.971−0.234i-0.971 - 0.234i
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 + 86.8i·5-s + 98.7i·7-s + 81·9-s + 610. i·11-s + (592. + 143. i)13-s + 781. i·15-s − 1.14e3·17-s + 2.26e3i·19-s + 888. i·21-s − 433.·23-s − 4.41e3·25-s + 729·27-s + 7.66e3·29-s + 7.36e3i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.55i·5-s + 0.761i·7-s + 0.333·9-s + 1.52i·11-s + (0.971 + 0.234i)13-s + 0.897i·15-s − 0.963·17-s + 1.44i·19-s + 0.439i·21-s − 0.170·23-s − 1.41·25-s + 0.192·27-s + 1.69·29-s + 1.37i·31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.971−0.234i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.234i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=((−0.971−0.234i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.971−0.234i-0.971 - 0.234i
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), −0.971−0.234i)(2,\ 624,\ (\ :5/2),\ -0.971 - 0.234i)

Particular Values

L(3)L(3) ≈\approx 2.6900190492.690019049
L(12)L(\frac12) ≈\approx 2.6900190492.690019049
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+(−592.−143.i)T 1 + (-592. - 143. i)T
good5 1−86.8iT−3.12e3T2 1 - 86.8iT - 3.12e3T^{2}
7 1−98.7iT−1.68e4T2 1 - 98.7iT - 1.68e4T^{2}
11 1−610.iT−1.61e5T2 1 - 610. iT - 1.61e5T^{2}
17 1+1.14e3T+1.41e6T2 1 + 1.14e3T + 1.41e6T^{2}
19 1−2.26e3iT−2.47e6T2 1 - 2.26e3iT - 2.47e6T^{2}
23 1+433.T+6.43e6T2 1 + 433.T + 6.43e6T^{2}
29 1−7.66e3T+2.05e7T2 1 - 7.66e3T + 2.05e7T^{2}
31 1−7.36e3iT−2.86e7T2 1 - 7.36e3iT - 2.86e7T^{2}
37 1+1.05e4iT−6.93e7T2 1 + 1.05e4iT - 6.93e7T^{2}
41 1−3.69e3iT−1.15e8T2 1 - 3.69e3iT - 1.15e8T^{2}
43 1−6.06e3T+1.47e8T2 1 - 6.06e3T + 1.47e8T^{2}
47 1−8.74e3iT−2.29e8T2 1 - 8.74e3iT - 2.29e8T^{2}
53 1−3.47e4T+4.18e8T2 1 - 3.47e4T + 4.18e8T^{2}
59 1+1.19e4iT−7.14e8T2 1 + 1.19e4iT - 7.14e8T^{2}
61 1+4.54e4T+8.44e8T2 1 + 4.54e4T + 8.44e8T^{2}
67 1+4.56e4iT−1.35e9T2 1 + 4.56e4iT - 1.35e9T^{2}
71 1+2.38e4iT−1.80e9T2 1 + 2.38e4iT - 1.80e9T^{2}
73 1+3.77e4iT−2.07e9T2 1 + 3.77e4iT - 2.07e9T^{2}
79 1−3.53e4T+3.07e9T2 1 - 3.53e4T + 3.07e9T^{2}
83 1−3.14e4iT−3.93e9T2 1 - 3.14e4iT - 3.93e9T^{2}
89 1+5.90e4iT−5.58e9T2 1 + 5.90e4iT - 5.58e9T^{2}
97 1+4.96e3iT−8.58e9T2 1 + 4.96e3iT - 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

−10.34502308784576475178183050912, −9.387018642100795803142021835039, −8.507688236771129014688361174740, −7.51174484146521779400399471214, −6.72659945489127875809895469565, −6.01070420502578545403821063739, −4.51850985886635876298616475890, −3.49548563156376494989870306763, −2.50460952834648100192678411711, −1.74006164358501563498803382170, 0.60399422954393106045835562205, 1.04640521067619181113254337760, 2.65351100516922038907593373695, 3.91239337289357224795388079396, 4.61017018053627682220870614476, 5.73156100919380872246432907298, 6.75869678639090112478506835928, 8.024821882266853532768251249942, 8.678219701210480503264542930186, 9.029098946886832169529376897368

Graph of the ZZ-function along the critical line