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Properties

Label 2-616-616.475-c0-0-1
Degree $2$
Conductor $616$
Sign $-0.605 + 0.795i$
Analytic cond. $0.307424$
Root an. cond. $0.554458$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(0.307424\)
Root analytic conductor: \(0.554458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :0),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6336524757\)
\(L(\frac12)\) \(\approx\) \(0.6336524757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line