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Properties

Label 2-416-416.99-c1-0-20
Degree $2$
Conductor $416$
Sign $0.190 - 0.981i$
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  + (−1.40 + 0.146i)2-s + (−0.514 + 1.24i)3-s + (1.95 − 0.411i)4-s + (−0.618 + 1.49i)5-s + (0.542 − 1.82i)6-s + 4.58·7-s + (−2.69 + 0.864i)8-s + (0.843 + 0.843i)9-s + (0.652 − 2.19i)10-s + (0.0872 + 0.0361i)11-s + (−0.496 + 2.64i)12-s + (0.783 − 3.51i)13-s + (−6.45 + 0.670i)14-s + (−1.53 − 1.53i)15-s + (3.66 − 1.60i)16-s + 2.55·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.103i)2-s + (−0.296 + 0.716i)3-s + (0.978 − 0.205i)4-s + (−0.276 + 0.668i)5-s + (0.221 − 0.743i)6-s + 1.73·7-s + (−0.952 + 0.305i)8-s + (0.281 + 0.281i)9-s + (0.206 − 0.693i)10-s + (0.0263 + 0.0108i)11-s + (−0.143 + 0.762i)12-s + (0.217 − 0.976i)13-s + (−1.72 + 0.179i)14-s + (−0.396 − 0.396i)15-s + (0.915 − 0.402i)16-s + 0.620·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.190 - 0.981i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749287 + 0.617793i\)
\(L(\frac12)\) \(\approx\) \(0.749287 + 0.617793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.146i)T \)
13 \( 1 + (-0.783 + 3.51i)T \)
good3 \( 1 + (0.514 - 1.24i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.618 - 1.49i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 - 4.58T + 7T^{2} \)
11 \( 1 + (-0.0872 - 0.0361i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 - 2.55T + 17T^{2} \)
19 \( 1 + (0.456 + 1.10i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.27 - 1.27i)T + 23iT^{2} \)
29 \( 1 + (-0.966 - 0.400i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.62 - 1.62i)T - 31iT^{2} \)
37 \( 1 + (-0.285 - 0.118i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 - 4.15iT - 41T^{2} \)
43 \( 1 + (8.33 - 3.45i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (6.32 - 6.32i)T - 47iT^{2} \)
53 \( 1 + (-3.50 + 1.45i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-5.30 + 12.8i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-7.13 - 2.95i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (13.6 - 5.66i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + 0.284iT - 71T^{2} \)
73 \( 1 - 1.57T + 73T^{2} \)
79 \( 1 + 3.48T + 79T^{2} \)
83 \( 1 + (2.91 + 7.04i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + (10.1 + 10.1i)T + 97iT^{2} \)
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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

−11.18179872767777914014034978087, −10.53357585549075223483653231109, −9.828850580537368417552386230697, −8.533095408710668098754533173597, −7.85427488044653708172902484389, −7.09030107794876489760638598041, −5.62450371911466080374409584828, −4.76450543754327888174678980288, −3.17702757026756942737211038519, −1.54338735709664151633885829392, 1.06542663758188521434020761208, 1.95791999781993410803455387247, 4.07134693487118298311564053588, 5.34320062218775638240695823749, 6.63663376773082590756317024190, 7.48935520477045794126026952953, 8.307584868350665478992367105087, 8.918221507796646419976981168430, 10.12682953622671667115268570225, 11.14913354408919950124928334847

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