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Properties

Label 2-416-104.77-c1-0-0
Degree $2$
Conductor $416$
Sign $-0.999 + 0.0422i$
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  + 1.51i·3-s − 3.11·5-s + 2.77i·7-s + 0.701·9-s − 2.56·11-s + (0.546 − 3.56i)13-s − 4.72i·15-s − 5.70·17-s − 4.75·19-s − 4.20·21-s − 4·23-s + 4.70·25-s + 5.61i·27-s + 7.12i·29-s − 3.60i·31-s + ⋯
L(s)  = 1  + 0.875i·3-s − 1.39·5-s + 1.04i·7-s + 0.233·9-s − 0.774·11-s + (0.151 − 0.988i)13-s − 1.21i·15-s − 1.38·17-s − 1.09·19-s − 0.918·21-s − 0.834·23-s + 0.940·25-s + 1.07i·27-s + 1.32i·29-s − 0.647i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.999 + 0.0422i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.999 + 0.0422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00954088 - 0.451410i\)
\(L(\frac12)\) \(\approx\) \(0.00954088 - 0.451410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.546 + 3.56i)T \)
good3 \( 1 - 1.51iT - 3T^{2} \)
5 \( 1 + 3.11T + 5T^{2} \)
7 \( 1 - 2.77iT - 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 7.12iT - 29T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 - 4.20T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 1.51iT - 43T^{2} \)
47 \( 1 - 2.77iT - 47T^{2} \)
53 \( 1 - 6.06iT - 53T^{2} \)
59 \( 1 - 4.75T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 6.66iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 - 3.89iT - 97T^{2} \)
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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

−11.43332133894879658915603482217, −10.86145268709529098196317800488, −9.969937570605474266841762647639, −8.775111340124169751029529479507, −8.234538318123397086050969764630, −7.16054389419975076306676080494, −5.78822876714233694420958032208, −4.68133403524991560639658881783, −3.89109038799450931534508184291, −2.61028954453465687991148734847, 0.27860535530311303967228990049, 2.12294246195311288318828431614, 3.98833159879635321498390163128, 4.45518042033171515715436621299, 6.42515758635716152455136733598, 7.09281301181701605072753737364, 7.85770488406135282657317130838, 8.552948380323674515706205412170, 10.01030025757259801049026503541, 10.97320298659395413937542608602

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