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Properties

Label 2-416-104.29-c1-0-7
Degree $2$
Conductor $416$
Sign $0.551 + 0.834i$
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  + (−0.609 + 0.351i)3-s − 2.24i·5-s + (−0.471 + 0.817i)7-s + (−1.25 + 2.16i)9-s + (4.82 − 2.78i)11-s + (−0.108 − 3.60i)13-s + (0.787 + 1.36i)15-s + (3.03 − 5.25i)17-s + (−1.43 − 0.831i)19-s − 0.663i·21-s + (−1.07 − 1.86i)23-s − 0.0179·25-s − 3.87i·27-s + (−1.68 + 0.972i)29-s + 1.91·31-s + ⋯
L(s)  = 1  + (−0.351 + 0.203i)3-s − 1.00i·5-s + (−0.178 + 0.308i)7-s + (−0.417 + 0.723i)9-s + (1.45 − 0.840i)11-s + (−0.0301 − 0.999i)13-s + (0.203 + 0.352i)15-s + (0.735 − 1.27i)17-s + (−0.330 − 0.190i)19-s − 0.144i·21-s + (−0.224 − 0.389i)23-s − 0.00358·25-s − 0.745i·27-s + (−0.312 + 0.180i)29-s + 0.343·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.551 + 0.834i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.551 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04391 - 0.561100i\)
\(L(\frac12)\) \(\approx\) \(1.04391 - 0.561100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.108 + 3.60i)T \)
good3 \( 1 + (0.609 - 0.351i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.24iT - 5T^{2} \)
7 \( 1 + (0.471 - 0.817i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.82 + 2.78i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.03 + 5.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.43 + 0.831i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.07 + 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.68 - 0.972i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.91T + 31T^{2} \)
37 \( 1 + (-4.54 + 2.62i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.332 + 0.575i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.97 + 4.02i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (0.411 + 0.237i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.06 + 1.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.91 - 2.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.81 - 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.91T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + (0.373 + 0.647i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.640 + 1.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19146104884440316602068794845, −10.17369914349466460116324408887, −9.085410192702872941905472864508, −8.560199539706069382698754414010, −7.44007394955316075734116334654, −6.02156642958499553943799672636, −5.36002265249506330521784397367, −4.30436388558094133619712977191, −2.85972415148528257020007278982, −0.882355883071305683608836266229, 1.65305835284773364808976851386, 3.40002541076206005694532394987, 4.26569923756187498655799826492, 6.04887138424777904972017619200, 6.57897787172943205007534061714, 7.34469576515371084310507330383, 8.721194017114553343930877523358, 9.664945394824406229143643304662, 10.43405365009268573487175977856, 11.57534717872681290238598234664

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