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Properties

Label 2-624-13.12-c5-0-43
Degree 22
Conductor 624624
Sign 0.910+0.413i0.910 + 0.413i
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 + 37.1i·5-s − 176. i·7-s + 81·9-s + 179. i·11-s + (−554. − 252. i)13-s + 334. i·15-s − 933.·17-s + 2.33e3i·19-s − 1.58e3i·21-s + 2.79e3·23-s + 1.74e3·25-s + 729·27-s + 1.50e3·29-s + 737. i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.663i·5-s − 1.36i·7-s + 0.333·9-s + 0.447i·11-s + (−0.910 − 0.413i)13-s + 0.383i·15-s − 0.783·17-s + 1.48i·19-s − 0.785i·21-s + 1.10·23-s + 0.559·25-s + 0.192·27-s + 0.331·29-s + 0.137i·31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((0.910+0.413i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=((0.910+0.413i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 0.910+0.413i0.910 + 0.413i
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), 0.910+0.413i)(2,\ 624,\ (\ :5/2),\ 0.910 + 0.413i)

Particular Values

L(3)L(3) ≈\approx 2.5208093412.520809341
L(12)L(\frac12) ≈\approx 2.5208093412.520809341
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+(554.+252.i)T 1 + (554. + 252. i)T
good5 1−37.1iT−3.12e3T2 1 - 37.1iT - 3.12e3T^{2}
7 1+176.iT−1.68e4T2 1 + 176. iT - 1.68e4T^{2}
11 1−179.iT−1.61e5T2 1 - 179. iT - 1.61e5T^{2}
17 1+933.T+1.41e6T2 1 + 933.T + 1.41e6T^{2}
19 1−2.33e3iT−2.47e6T2 1 - 2.33e3iT - 2.47e6T^{2}
23 1−2.79e3T+6.43e6T2 1 - 2.79e3T + 6.43e6T^{2}
29 1−1.50e3T+2.05e7T2 1 - 1.50e3T + 2.05e7T^{2}
31 1−737.iT−2.86e7T2 1 - 737. iT - 2.86e7T^{2}
37 1+3.77e3iT−6.93e7T2 1 + 3.77e3iT - 6.93e7T^{2}
41 1−368.iT−1.15e8T2 1 - 368. iT - 1.15e8T^{2}
43 1−2.01e4T+1.47e8T2 1 - 2.01e4T + 1.47e8T^{2}
47 1+2.05e4iT−2.29e8T2 1 + 2.05e4iT - 2.29e8T^{2}
53 1+2.50e4T+4.18e8T2 1 + 2.50e4T + 4.18e8T^{2}
59 1+3.53e4iT−7.14e8T2 1 + 3.53e4iT - 7.14e8T^{2}
61 1−3.17e4T+8.44e8T2 1 - 3.17e4T + 8.44e8T^{2}
67 1−4.66e4iT−1.35e9T2 1 - 4.66e4iT - 1.35e9T^{2}
71 1+5.89e4iT−1.80e9T2 1 + 5.89e4iT - 1.80e9T^{2}
73 1−3.41e3iT−2.07e9T2 1 - 3.41e3iT - 2.07e9T^{2}
79 1−6.49e4T+3.07e9T2 1 - 6.49e4T + 3.07e9T^{2}
83 1+1.22e4iT−3.93e9T2 1 + 1.22e4iT - 3.93e9T^{2}
89 1+6.17e4iT−5.58e9T2 1 + 6.17e4iT - 5.58e9T^{2}
97 1+2.80e4iT−8.58e9T2 1 + 2.80e4iT - 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

−9.994321990751631726211909834170, −8.934646620525931159010058621146, −7.77719226677921782587794991850, −7.25478744652236884443160063417, −6.48934260075849633188225203430, −5.00046622963145585683443603479, −4.03331587516143746352458234440, −3.12890028179350542320648533678, −1.99689509381207852773075736137, −0.64610803080619483784410394620, 0.845292682097967014254869906180, 2.30475672189947775061111889174, 2.90709900838358671446565450491, 4.53171258024345447444663419704, 5.12671468352420220442452956881, 6.32710787222062382506275000394, 7.29833141133885001936709906848, 8.441244608034686328082594893188, 9.097536459028323040203458296051, 9.382260199361189844657791496096

Graph of the ZZ-function along the critical line