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Properties

Label 2-416-416.83-c1-0-29
Degree $2$
Conductor $416$
Sign $0.904 + 0.425i$
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  + (−1.01 − 0.985i)2-s + (2.77 + 1.14i)3-s + (0.0589 + 1.99i)4-s + (0.336 − 0.813i)5-s + (−1.68 − 3.89i)6-s − 2.04i·7-s + (1.90 − 2.08i)8-s + (4.25 + 4.25i)9-s + (−1.14 + 0.493i)10-s + (2.41 + 1.00i)11-s + (−2.13 + 5.61i)12-s + (−0.624 − 3.55i)13-s + (−2.01 + 2.07i)14-s + (1.87 − 1.87i)15-s + (−3.99 + 0.235i)16-s − 1.32·17-s + ⋯
L(s)  = 1  + (−0.717 − 0.696i)2-s + (1.60 + 0.663i)3-s + (0.0294 + 0.999i)4-s + (0.150 − 0.363i)5-s + (−0.687 − 1.59i)6-s − 0.773i·7-s + (0.675 − 0.737i)8-s + (1.41 + 1.41i)9-s + (−0.361 + 0.156i)10-s + (0.729 + 0.302i)11-s + (−0.616 + 1.62i)12-s + (−0.173 − 0.984i)13-s + (−0.538 + 0.554i)14-s + (0.482 − 0.482i)15-s + (−0.998 + 0.0589i)16-s − 0.321·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.904 + 0.425i$
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: \((2,\ 416,\ (\ :1/2),\ 0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66473 - 0.372151i\)
\(L(\frac12)\) \(\approx\) \(1.66473 - 0.372151i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 + 0.985i)T \)
13 \( 1 + (0.624 + 3.55i)T \)
good3 \( 1 + (-2.77 - 1.14i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.336 + 0.813i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + 2.04iT - 7T^{2} \)
11 \( 1 + (-2.41 - 1.00i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 + (0.0902 + 0.217i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.792 + 0.792i)T + 23iT^{2} \)
29 \( 1 + (3.41 - 8.24i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.40 - 1.40i)T + 31iT^{2} \)
37 \( 1 + (-4.08 - 1.69i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + 5.02T + 41T^{2} \)
43 \( 1 + (2.93 + 7.07i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (1.26 + 1.26i)T + 47iT^{2} \)
53 \( 1 + (2.71 + 6.56i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.13 - 9.99i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.03 - 2.49i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (10.2 - 4.22i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + (3.08 + 7.44i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 0.488T + 89T^{2} \)
97 \( 1 + (-6.29 + 6.29i)T - 97iT^{2} \)
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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

−10.60683859080371466283392166885, −10.22832064150290387223999668848, −9.172024830130982291223043799432, −8.790725765558674122938110225670, −7.78831976139134395970395206541, −7.03181810512479518795880563510, −4.79311243678165605042139880267, −3.78697945371717753169359300329, −2.97529819308040951026614540926, −1.55251409348233025658785068681, 1.71073900912839909209160363116, 2.69310325319326056049352212828, 4.29747367945995795469494361303, 6.10419017030279556205330285374, 6.77306277817138784819405230126, 7.77040562504490243544859618333, 8.506127429434668789688038410710, 9.264431142005957855692848325760, 9.735554841542310672460595860465, 11.20372427542711549227738682996

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