Properties

Label 2-416-8.5-c1-0-10
Degree $2$
Conductor $416$
Sign $-0.707 + 0.707i$
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  i·3-s − 3i·5-s − 3·7-s + 2·9-s i·13-s − 3·15-s − 7·17-s − 4i·19-s + 3i·21-s − 4·23-s − 4·25-s − 5i·27-s + 4i·29-s + 8·31-s + 9i·35-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.34i·5-s − 1.13·7-s + 0.666·9-s − 0.277i·13-s − 0.774·15-s − 1.69·17-s − 0.917i·19-s + 0.654i·21-s − 0.834·23-s − 0.800·25-s − 0.962i·27-s + 0.742i·29-s + 1.43·31-s + 1.52i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 416,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.389250 - 0.939732i\)
\(L(\frac12)\) \(\approx\) \(0.389250 - 0.939732i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 7T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.88697112495268797091923845999, −9.752886466224685561337253962274, −9.067145886263812370717231823172, −8.212080707638445255604324338111, −7.00977517809252333308656063861, −6.29294867830547392975854592743, −4.96686704061183388838479542014, −4.01260748526645672830388922689, −2.29119443694623589441939074338, −0.64360083432287352970178500465, 2.41091872202386334518782012665, 3.56791279260911188479866218575, 4.46974671565504062997968650541, 6.25777464968427212796214593142, 6.61711645796808404555812741294, 7.73026060178007962197194849978, 9.089775066662220953126225337654, 10.01564076271265384887889462698, 10.38826353783580439289590554285, 11.36169354213970672628326571399

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