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Properties

Label 2-357-119.10-c1-0-17
Degree 22
Conductor 357357
Sign 0.181+0.983i0.181 + 0.983i
Analytic cond. 2.850652.85065
Root an. cond. 1.688381.68838
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  + (0.559 − 0.0736i)2-s + (−0.442 − 0.896i)3-s + (−1.62 + 0.435i)4-s + (1.94 − 1.70i)5-s + (−0.313 − 0.469i)6-s + (2.53 − 0.746i)7-s + (−1.91 + 0.794i)8-s + (−0.608 + 0.793i)9-s + (0.962 − 1.09i)10-s + (0.958 − 0.0627i)11-s + (1.10 + 1.26i)12-s + (−3.77 − 3.77i)13-s + (1.36 − 0.604i)14-s + (−2.39 − 0.990i)15-s + (1.89 − 1.09i)16-s + (3.07 − 2.74i)17-s + ⋯
L(s)  = 1  + (0.395 − 0.0520i)2-s + (−0.255 − 0.517i)3-s + (−0.812 + 0.217i)4-s + (0.870 − 0.763i)5-s + (−0.127 − 0.191i)6-s + (0.959 − 0.282i)7-s + (−0.678 + 0.280i)8-s + (−0.202 + 0.264i)9-s + (0.304 − 0.347i)10-s + (0.288 − 0.0189i)11-s + (0.320 + 0.365i)12-s + (−1.04 − 1.04i)13-s + (0.364 − 0.161i)14-s + (−0.617 − 0.255i)15-s + (0.474 − 0.274i)16-s + (0.745 − 0.666i)17-s + ⋯

Functional equation

Λ(s)=(357s/2 ΓC(s) L(s)=((0.181+0.983i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(357s/2 ΓC(s+1/2) L(s)=((0.181+0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 357357    =    3⋅7⋅173 \cdot 7 \cdot 17
Sign: 0.181+0.983i0.181 + 0.983i
Analytic conductor: 2.850652.85065
Root analytic conductor: 1.688381.68838
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ357(10,⋅)\chi_{357} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 357, ( :1/2), 0.181+0.983i)(2,\ 357,\ (\ :1/2),\ 0.181 + 0.983i)

Particular Values

L(1)L(1) ≈\approx 1.10729−0.921882i1.10729 - 0.921882i
L(12)L(\frac12) ≈\approx 1.10729−0.921882i1.10729 - 0.921882i
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)
bad3 1+(0.442+0.896i)T 1 + (0.442 + 0.896i)T
7 1+(−2.53+0.746i)T 1 + (-2.53 + 0.746i)T
17 1+(−3.07+2.74i)T 1 + (-3.07 + 2.74i)T
good2 1+(−0.559+0.0736i)T+(1.93−0.517i)T2 1 + (-0.559 + 0.0736i)T + (1.93 - 0.517i)T^{2}
5 1+(−1.94+1.70i)T+(0.652−4.95i)T2 1 + (-1.94 + 1.70i)T + (0.652 - 4.95i)T^{2}
11 1+(−0.958+0.0627i)T+(10.9−1.43i)T2 1 + (-0.958 + 0.0627i)T + (10.9 - 1.43i)T^{2}
13 1+(3.77+3.77i)T+13iT2 1 + (3.77 + 3.77i)T + 13iT^{2}
19 1+(0.417+3.16i)T+(−18.3+4.91i)T2 1 + (0.417 + 3.16i)T + (-18.3 + 4.91i)T^{2}
23 1+(3.58+1.76i)T+(14.0+18.2i)T2 1 + (3.58 + 1.76i)T + (14.0 + 18.2i)T^{2}
29 1+(−0.353−1.77i)T+(−26.7+11.0i)T2 1 + (-0.353 - 1.77i)T + (-26.7 + 11.0i)T^{2}
31 1+(−9.23+4.55i)T+(18.8−24.5i)T2 1 + (-9.23 + 4.55i)T + (18.8 - 24.5i)T^{2}
37 1+(0.606−9.26i)T+(−36.6−4.82i)T2 1 + (0.606 - 9.26i)T + (-36.6 - 4.82i)T^{2}
41 1+(1.08−5.44i)T+(−37.8−15.6i)T2 1 + (1.08 - 5.44i)T + (-37.8 - 15.6i)T^{2}
43 1+(0.368+0.889i)T+(−30.4+30.4i)T2 1 + (0.368 + 0.889i)T + (-30.4 + 30.4i)T^{2}
47 1+(3.22−12.0i)T+(−40.7−23.5i)T2 1 + (3.22 - 12.0i)T + (-40.7 - 23.5i)T^{2}
53 1+(−0.111−0.145i)T+(−13.7+51.1i)T2 1 + (-0.111 - 0.145i)T + (-13.7 + 51.1i)T^{2}
59 1+(11.5+1.52i)T+(56.9+15.2i)T2 1 + (11.5 + 1.52i)T + (56.9 + 15.2i)T^{2}
61 1+(−3.35−9.88i)T+(−48.3+37.1i)T2 1 + (-3.35 - 9.88i)T + (-48.3 + 37.1i)T^{2}
67 1+(−5.04−2.91i)T+(33.5+58.0i)T2 1 + (-5.04 - 2.91i)T + (33.5 + 58.0i)T^{2}
71 1+(3.65+2.44i)T+(27.1+65.5i)T2 1 + (3.65 + 2.44i)T + (27.1 + 65.5i)T^{2}
73 1+(−9.25−3.14i)T+(57.9+44.4i)T2 1 + (-9.25 - 3.14i)T + (57.9 + 44.4i)T^{2}
79 1+(−6.43+13.0i)T+(−48.0−62.6i)T2 1 + (-6.43 + 13.0i)T + (-48.0 - 62.6i)T^{2}
83 1+(6.07−14.6i)T+(−58.6−58.6i)T2 1 + (6.07 - 14.6i)T + (-58.6 - 58.6i)T^{2}
89 1+(−9.93−2.66i)T+(77.0+44.5i)T2 1 + (-9.93 - 2.66i)T + (77.0 + 44.5i)T^{2}
97 1+(−5.25+1.04i)T+(89.6−37.1i)T2 1 + (-5.25 + 1.04i)T + (89.6 - 37.1i)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

−11.60857104225244900657931502432, −10.17289402493474488223940943242, −9.459470402731925844363295042295, −8.324045269913900486189297029993, −7.68932171743920569974096855866, −6.14547927587188771535406072842, −5.08415468591289071251532296971, −4.66486536470864084842367950088, −2.76253493598468979025076756122, −1.01014758083278968851419526848, 2.01588348733456223427438672156, 3.73906418882047566434573926080, 4.79490862287624218744694170600, 5.66509447209820807214450730586, 6.53834034088505474039715653807, 8.014080503632672113521244155285, 9.104387982551011987845305492287, 9.948382059863666150426342503709, 10.48350286034594897677555224012, 11.81370819426971325553161795270

Graph of the ZZ-function along the critical line