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Properties

Label 2-416-416.83-c1-0-23
Degree $2$
Conductor $416$
Sign $0.843 - 0.537i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 0.0764i)2-s + (2.23 + 0.925i)3-s + (1.98 + 0.215i)4-s + (0.344 − 0.832i)5-s + (−3.08 − 1.47i)6-s + 1.15i·7-s + (−2.79 − 0.456i)8-s + (2.01 + 2.01i)9-s + (−0.550 + 1.14i)10-s + (0.739 + 0.306i)11-s + (4.24 + 2.32i)12-s + (2.43 + 2.66i)13-s + (0.0881 − 1.62i)14-s + (1.54 − 1.54i)15-s + (3.90 + 0.858i)16-s + 2.50·17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0540i)2-s + (1.29 + 0.534i)3-s + (0.994 + 0.107i)4-s + (0.154 − 0.372i)5-s + (−1.25 − 0.603i)6-s + 0.436i·7-s + (−0.986 − 0.161i)8-s + (0.672 + 0.672i)9-s + (−0.174 + 0.363i)10-s + (0.223 + 0.0923i)11-s + (1.22 + 0.670i)12-s + (0.674 + 0.738i)13-s + (0.0235 − 0.435i)14-s + (0.398 − 0.398i)15-s + (0.976 + 0.214i)16-s + 0.608·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.843 - 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38484 + 0.403970i\)
\(L(\frac12)\) \(\approx\) \(1.38484 + 0.403970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0764i)T \)
13 \( 1 + (-2.43 - 2.66i)T \)
good3 \( 1 + (-2.23 - 0.925i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.344 + 0.832i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 - 1.15iT - 7T^{2} \)
11 \( 1 + (-0.739 - 0.306i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 - 2.50T + 17T^{2} \)
19 \( 1 + (-0.0724 - 0.174i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.04 + 2.04i)T + 23iT^{2} \)
29 \( 1 + (-0.0118 + 0.0286i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.77 + 1.77i)T + 31iT^{2} \)
37 \( 1 + (6.94 + 2.87i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 - 3.61T + 41T^{2} \)
43 \( 1 + (-2.96 - 7.16i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (4.85 + 4.85i)T + 47iT^{2} \)
53 \( 1 + (0.104 + 0.252i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.43 + 5.87i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.846 + 2.04i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.88 - 2.43i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + 1.39T + 71T^{2} \)
73 \( 1 - 0.899iT - 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (-2.03 - 4.91i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + (-5.50 + 5.50i)T - 97iT^{2} \)
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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

−11.06098903600795722765893939465, −10.00709601022633194921184773581, −9.286119030705264446838942514251, −8.743673526176044349905954879870, −8.051543455928785973505765619513, −6.92467690021518798310173868276, −5.68960182627826131900352753535, −4.03895844981956036933198987030, −2.93633937765982102462812193849, −1.70342412002524526777304122515, 1.34269039740878937388025193794, 2.69009417942270843168434254420, 3.59675528952338096093944233281, 5.74729017460811676305622672239, 6.90070552250154846257954613369, 7.62023878936095320831219871729, 8.403733178347679350882493053255, 9.079468971185922194615789250832, 10.14839925616449481012838609240, 10.76907528554740042533475026299

Graph of the $Z$-function along the critical line