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Properties

Label 2-416-13.3-c1-0-5
Degree $2$
Conductor $416$
Sign $0.0128 - 0.999i$
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  + (1.65 + 2.87i)3-s + (1.65 − 2.87i)7-s + (−4 + 6.92i)9-s + (1.65 + 2.87i)11-s + (−1 + 3.46i)13-s + (1.5 − 2.59i)17-s + (1.65 − 2.87i)19-s + 11·21-s + (−1.65 − 2.87i)23-s − 5·25-s − 16.5·27-s + (2.5 + 4.33i)29-s + (−5.5 + 9.52i)33-s + (−4.5 − 7.79i)37-s + (−11.6 + 2.87i)39-s + ⋯
L(s)  = 1  + (0.957 + 1.65i)3-s + (0.626 − 1.08i)7-s + (−1.33 + 2.30i)9-s + (0.500 + 0.866i)11-s + (−0.277 + 0.960i)13-s + (0.363 − 0.630i)17-s + (0.380 − 0.658i)19-s + 2.40·21-s + (−0.345 − 0.598i)23-s − 25-s − 3.19·27-s + (0.464 + 0.804i)29-s + (−0.957 + 1.65i)33-s + (−0.739 − 1.28i)37-s + (−1.85 + 0.459i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36400 + 1.34662i\)
\(L(\frac12)\) \(\approx\) \(1.36400 + 1.34662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (-1.65 - 2.87i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (-1.65 + 2.87i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.65 - 2.87i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.65 + 2.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.65 + 2.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (4.5 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.97 + 8.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.63T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (-1.65 + 2.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.97 - 8.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.97 + 8.61i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 6.63T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.08393222630473618117997142836, −10.41146494491583582180983605036, −9.555010468223417486345730071864, −9.035589203665678882415639145963, −7.87273121172963451066923323974, −7.05029437252841768071911368005, −5.15945733075600577077676271597, −4.38431073096514722446644601749, −3.73067982323631693501141951886, −2.21698153858537655787529738817, 1.31218675849413734944107603764, 2.48349446992287228183830790898, 3.53363272033310291419500223737, 5.66840111213850985847655353908, 6.20630909665338915881201577156, 7.60955100485619283118037922508, 8.137168737027138719099948848284, 8.760104854956895070987330381274, 9.834054597293179805332052482999, 11.43994153228692479353828043939

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