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Properties

Label 2-527-31.4-c1-0-29
Degree 22
Conductor 527527
Sign 0.971−0.238i0.971 - 0.238i
Analytic cond. 4.208114.20811
Root an. cond. 2.051362.05136
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0573 + 0.176i)2-s + (−0.417 + 1.28i)3-s + (1.59 − 1.15i)4-s + 2.24·5-s − 0.250·6-s + (2.41 − 1.75i)7-s + (0.595 + 0.432i)8-s + (0.953 + 0.692i)9-s + (0.128 + 0.396i)10-s + (−4.52 + 3.28i)11-s + (0.819 + 2.52i)12-s + (1.77 − 5.45i)13-s + (0.449 + 0.326i)14-s + (−0.936 + 2.88i)15-s + (1.17 − 3.60i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.0405 + 0.124i)2-s + (−0.240 + 0.741i)3-s + (0.795 − 0.577i)4-s + 1.00·5-s − 0.102·6-s + (0.914 − 0.664i)7-s + (0.210 + 0.152i)8-s + (0.317 + 0.230i)9-s + (0.0407 + 0.125i)10-s + (−1.36 + 0.990i)11-s + (0.236 + 0.728i)12-s + (0.491 − 1.51i)13-s + (0.120 + 0.0872i)14-s + (−0.241 + 0.744i)15-s + (0.293 − 0.902i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

Λ(s)=(527s/2 ΓC(s) L(s)=((0.971−0.238i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(527s/2 ΓC(s+1/2) L(s)=((0.971−0.238i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 527527    =    17⋅3117 \cdot 31
Sign: 0.971−0.238i0.971 - 0.238i
Analytic conductor: 4.208114.20811
Root analytic conductor: 2.051362.05136
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ527(35,⋅)\chi_{527} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 527, ( :1/2), 0.971−0.238i)(2,\ 527,\ (\ :1/2),\ 0.971 - 0.238i)

Particular Values

L(1)L(1) ≈\approx 2.02427+0.244587i2.02427 + 0.244587i
L(12)L(\frac12) ≈\approx 2.02427+0.244587i2.02427 + 0.244587i
L(32)L(\frac{3}{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)
bad17 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
31 1+(5.56+0.200i)T 1 + (5.56 + 0.200i)T
good2 1+(−0.0573−0.176i)T+(−1.61+1.17i)T2 1 + (-0.0573 - 0.176i)T + (-1.61 + 1.17i)T^{2}
3 1+(0.417−1.28i)T+(−2.42−1.76i)T2 1 + (0.417 - 1.28i)T + (-2.42 - 1.76i)T^{2}
5 1−2.24T+5T2 1 - 2.24T + 5T^{2}
7 1+(−2.41+1.75i)T+(2.16−6.65i)T2 1 + (-2.41 + 1.75i)T + (2.16 - 6.65i)T^{2}
11 1+(4.52−3.28i)T+(3.39−10.4i)T2 1 + (4.52 - 3.28i)T + (3.39 - 10.4i)T^{2}
13 1+(−1.77+5.45i)T+(−10.5−7.64i)T2 1 + (-1.77 + 5.45i)T + (-10.5 - 7.64i)T^{2}
19 1+(0.737+2.27i)T+(−15.3+11.1i)T2 1 + (0.737 + 2.27i)T + (-15.3 + 11.1i)T^{2}
23 1+(1.65+1.20i)T+(7.10+21.8i)T2 1 + (1.65 + 1.20i)T + (7.10 + 21.8i)T^{2}
29 1+(−1.74−5.35i)T+(−23.4+17.0i)T2 1 + (-1.74 - 5.35i)T + (-23.4 + 17.0i)T^{2}
37 1−4.55T+37T2 1 - 4.55T + 37T^{2}
41 1+(−1.55−4.79i)T+(−33.1+24.0i)T2 1 + (-1.55 - 4.79i)T + (-33.1 + 24.0i)T^{2}
43 1+(−2.97−9.16i)T+(−34.7+25.2i)T2 1 + (-2.97 - 9.16i)T + (-34.7 + 25.2i)T^{2}
47 1+(2.06−6.34i)T+(−38.0−27.6i)T2 1 + (2.06 - 6.34i)T + (-38.0 - 27.6i)T^{2}
53 1+(2.30+1.67i)T+(16.3+50.4i)T2 1 + (2.30 + 1.67i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.552+1.69i)T+(−47.7−34.6i)T2 1 + (-0.552 + 1.69i)T + (-47.7 - 34.6i)T^{2}
61 1−3.77T+61T2 1 - 3.77T + 61T^{2}
67 1−5.18T+67T2 1 - 5.18T + 67T^{2}
71 1+(12.5+9.10i)T+(21.9+67.5i)T2 1 + (12.5 + 9.10i)T + (21.9 + 67.5i)T^{2}
73 1+(10.5−7.63i)T+(22.5−69.4i)T2 1 + (10.5 - 7.63i)T + (22.5 - 69.4i)T^{2}
79 1+(0.758+0.551i)T+(24.4+75.1i)T2 1 + (0.758 + 0.551i)T + (24.4 + 75.1i)T^{2}
83 1+(1.29+3.97i)T+(−67.1+48.7i)T2 1 + (1.29 + 3.97i)T + (-67.1 + 48.7i)T^{2}
89 1+(−11.6+8.49i)T+(27.5−84.6i)T2 1 + (-11.6 + 8.49i)T + (27.5 - 84.6i)T^{2}
97 1+(0.642−0.466i)T+(29.9−92.2i)T2 1 + (0.642 - 0.466i)T + (29.9 - 92.2i)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

−10.66433293049378812228480250907, −10.26906427279673849500874999570, −9.546668299270770460370700444927, −7.929057160672936978820432484227, −7.37941728036486641987435688524, −6.05145968071330166627094466768, −5.20415307512504645897847233671, −4.62875298595070656663145961613, −2.72906754982018369325501236581, −1.54239977956998992754757292730, 1.75508451378149840373344474739, 2.32090016091787389824411956478, 3.97695940738487493900274656401, 5.60645751977636580138983623971, 6.14129762848257550075387373534, 7.17113763050249977280918737428, 8.068662689596221231073143683930, 8.864427958006547403103007767828, 10.09750036258226685552818443205, 11.07190489420973985892743584557

Graph of the ZZ-function along the critical line