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Properties

Label 2-812-812.503-c0-0-1
Degree 22
Conductor 812812
Sign −0.521+0.853i-0.521 + 0.853i
Analytic cond. 0.4052400.405240
Root an. cond. 0.6365850.636585
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.433 − 0.900i)7-s + (−0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (0.222 + 0.0250i)11-s + (−0.974 − 0.222i)14-s + (−0.900 + 0.433i)16-s + (−0.974 + 0.222i)18-s + (0.158 − 0.158i)22-s + (1.75 − 0.400i)23-s + (0.900 + 0.433i)25-s + (−0.781 + 0.623i)28-s + (−0.433 + 0.900i)29-s + (−0.222 + 0.974i)32-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.433 − 0.900i)7-s + (−0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (0.222 + 0.0250i)11-s + (−0.974 − 0.222i)14-s + (−0.900 + 0.433i)16-s + (−0.974 + 0.222i)18-s + (0.158 − 0.158i)22-s + (1.75 − 0.400i)23-s + (0.900 + 0.433i)25-s + (−0.781 + 0.623i)28-s + (−0.433 + 0.900i)29-s + (−0.222 + 0.974i)32-s + ⋯

Functional equation

Λ(s)=(812s/2 ΓC(s) L(s)=((−0.521+0.853i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(812s/2 ΓC(s) L(s)=((−0.521+0.853i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 812812    =    22⋅7⋅292^{2} \cdot 7 \cdot 29
Sign: −0.521+0.853i-0.521 + 0.853i
Analytic conductor: 0.4052400.405240
Root analytic conductor: 0.6365850.636585
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ812(503,⋅)\chi_{812} (503, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 812, ( :0), −0.521+0.853i)(2,\ 812,\ (\ :0),\ -0.521 + 0.853i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1492121671.149212167
L(12)L(\frac12) ≈\approx 1.1492121671.149212167
L(1)L(1) 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+(−0.623+0.781i)T 1 + (-0.623 + 0.781i)T
7 1+(0.433+0.900i)T 1 + (0.433 + 0.900i)T
29 1+(0.433−0.900i)T 1 + (0.433 - 0.900i)T
good3 1+(0.781+0.623i)T2 1 + (0.781 + 0.623i)T^{2}
5 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
11 1+(−0.222−0.0250i)T+(0.974+0.222i)T2 1 + (-0.222 - 0.0250i)T + (0.974 + 0.222i)T^{2}
13 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+(−0.781+0.623i)T2 1 + (-0.781 + 0.623i)T^{2}
23 1+(−1.75+0.400i)T+(0.900−0.433i)T2 1 + (-1.75 + 0.400i)T + (0.900 - 0.433i)T^{2}
31 1+(0.433−0.900i)T2 1 + (0.433 - 0.900i)T^{2}
37 1+(0.189+1.68i)T+(−0.974+0.222i)T2 1 + (0.189 + 1.68i)T + (-0.974 + 0.222i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(−1.19−0.752i)T+(0.433+0.900i)T2 1 + (-1.19 - 0.752i)T + (0.433 + 0.900i)T^{2}
47 1+(0.974+0.222i)T2 1 + (0.974 + 0.222i)T^{2}
53 1+(0.277−1.21i)T+(−0.900−0.433i)T2 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2}
59 1+T2 1 + T^{2}
61 1+(−0.781−0.623i)T2 1 + (-0.781 - 0.623i)T^{2}
67 1+(0.541−0.678i)T+(−0.222−0.974i)T2 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2}
71 1+(−0.277−0.347i)T+(−0.222+0.974i)T2 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2}
73 1+(−0.433−0.900i)T2 1 + (-0.433 - 0.900i)T^{2}
79 1+(0.222+1.97i)T+(−0.974+0.222i)T2 1 + (0.222 + 1.97i)T + (-0.974 + 0.222i)T^{2}
83 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
89 1+(−0.433+0.900i)T2 1 + (-0.433 + 0.900i)T^{2}
97 1+(−0.781+0.623i)T2 1 + (-0.781 + 0.623i)T^{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

−10.54996357261380118268460009511, −9.185674311701639375300079966077, −9.044339732729913238604242935263, −7.36401719895463302437793063088, −6.56642839174760492367994790459, −5.62570282726942069021743606919, −4.59469017798466834568313167181, −3.56773592863378161282438772407, −2.81271668400874590573770523835, −1.03977187993041175460159173928, 2.51936981496033461507104549681, 3.34025302566291355020819175693, 4.75809204178692826791180773183, 5.45856186950826579814149741248, 6.30225833664594189833148158733, 7.13891165424835747891618727983, 8.230676782181550812764760543735, 8.796061156247765358998787011917, 9.617335506014708386630449780734, 10.97877910620312294930708160764

Graph of the ZZ-function along the critical line