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Properties

Label 2-616-616.475-c0-0-1
Degree 22
Conductor 616616
Sign −0.605+0.795i-0.605 + 0.795i
Analytic cond. 0.3074240.307424
Root an. cond. 0.5544580.554458
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s − 0.999·18-s + (−0.809 + 0.587i)22-s − 1.17i·23-s + (0.809 − 0.587i)25-s + 0.999·28-s + (−0.5 − 0.363i)29-s − 32-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s − 0.999·18-s + (−0.809 + 0.587i)22-s − 1.17i·23-s + (0.809 − 0.587i)25-s + 0.999·28-s + (−0.5 − 0.363i)29-s − 32-s + ⋯

Functional equation

Λ(s)=(616s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(616s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    23⋅7⋅112^{3} \cdot 7 \cdot 11
Sign: −0.605+0.795i-0.605 + 0.795i
Analytic conductor: 0.3074240.307424
Root analytic conductor: 0.5544580.554458
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ616(475,⋅)\chi_{616} (475, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 616, ( :0), −0.605+0.795i)(2,\ 616,\ (\ :0),\ -0.605 + 0.795i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.63365247570.6336524757
L(12)L(\frac12) ≈\approx 0.63365247570.6336524757
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+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
7 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
11 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
good3 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
5 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
13 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
17 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
19 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
23 1+1.17iT−T2 1 + 1.17iT - T^{2}
29 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
31 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
37 1+(0.690−0.951i)T+(−0.309−0.951i)T2 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1−1.90iT−T2 1 - 1.90iT - T^{2}
47 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
53 1+(−1.80−0.587i)T+(0.809+0.587i)T2 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
67 1+0.618T+T2 1 + 0.618T + T^{2}
71 1+(−1.80+0.587i)T+(0.809−0.587i)T2 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2}
73 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
79 1+(0.190−0.587i)T+(−0.809−0.587i)T2 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}
83 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)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

−10.46990422625143956994409073863, −9.831649941991043834523976619938, −8.963067586437444394392553396201, −8.190065140699291241794142552370, −6.99722605450992703747572544144, −6.08174767177205357595525021400, −4.60020953849924232237224761952, −3.61862836859801948442511684336, −2.77376311671252114002137359776, −0.849792243372756484325123285709, 2.03572329521647200927007248431, 3.75287951276150827112869079563, 5.08791934335654805435240970050, 5.62111622194256287673052976618, 6.99753379381835989336756536764, 7.36194963039937248938043260493, 8.531351090365129666374990789024, 9.311173752890011606928179974763, 10.07435153527039906797812953343, 10.80828203363443239986681050097

Graph of the ZZ-function along the critical line