Properties

Label 2-416-8.5-c1-0-5
Degree $2$
Conductor $416$
Sign $-0.258 - 0.965i$
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  + 2i·3-s + 3.46i·5-s + 4.73·7-s − 9-s − 1.26i·11-s i·13-s − 6.92·15-s − 1.46·17-s − 2.73i·19-s + 9.46i·21-s − 4·23-s − 6.99·25-s + 4i·27-s − 2i·29-s + 3.26·31-s + ⋯
L(s)  = 1  + 1.15i·3-s + 1.54i·5-s + 1.78·7-s − 0.333·9-s − 0.382i·11-s − 0.277i·13-s − 1.78·15-s − 0.355·17-s − 0.626i·19-s + 2.06i·21-s − 0.834·23-s − 1.39·25-s + 0.769i·27-s − 0.371i·29-s + 0.586·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.987792 + 1.28731i\)
\(L(\frac12)\) \(\approx\) \(0.987792 + 1.28731i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 + 1.26iT - 11T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 3.26T + 31T^{2} \)
37 \( 1 + 4.92iT - 37T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 - 0.196iT - 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 + 2.73iT - 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 0.535T + 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 6.73iT - 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−11.18347939226068674769171128578, −10.64073027091680827405028239155, −9.998905142741516134238077626976, −8.758831745200813004232375446695, −7.83355508684672240742556567360, −6.86174036002332583599412977952, −5.55997228536087349195201798109, −4.56341897588010696309581687180, −3.57585239248970004918043916706, −2.22449239575256709356198648105, 1.25787348341261807773421410633, 1.91365119986622245651772896758, 4.38201926300298958803600849563, 4.98462993962920393861146330572, 6.19942722935809631447847378801, 7.53917559313879621791417810081, 8.131188689436848486045260496386, 8.744082584547826445155678671743, 9.972408185142455694800867205414, 11.31612240228864407876460987867

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