Properties

Label 2-416-13.12-c1-0-1
Degree $2$
Conductor $416$
Sign $-i$
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  + 2.60i·7-s − 3·9-s + 6.60i·11-s − 3.60·13-s + 7.21·17-s − 1.39i·19-s + 5·25-s − 7.21·29-s + 10.6i·31-s − 5.39i·47-s + 0.211·49-s − 2·53-s − 11.8i·59-s + 6·61-s − 7.81i·63-s + ⋯
L(s)  = 1  + 0.984i·7-s − 9-s + 1.99i·11-s − 1.00·13-s + 1.74·17-s − 0.319i·19-s + 25-s − 1.33·29-s + 1.90i·31-s − 0.786i·47-s + 0.0301·49-s − 0.274·53-s − 1.53i·59-s + 0.768·61-s − 0.984i·63-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.774458 + 0.774458i\)
\(L(\frac12)\) \(\approx\) \(0.774458 + 0.774458i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.60T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 2.60iT - 7T^{2} \)
11 \( 1 - 6.60iT - 11T^{2} \)
17 \( 1 - 7.21T + 17T^{2} \)
19 \( 1 + 1.39iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 7.21T + 29T^{2} \)
31 \( 1 - 10.6iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 5.39iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 11.8iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 14.6iT - 67T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 3.81iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.69329853030028577295600433285, −10.39183781127004033472534018583, −9.608050738002508498892501317916, −8.820540867512384582929234010368, −7.69513306712950085322864099729, −6.87136780288255110084585463440, −5.46692934704406891174584627231, −4.91638418915529107947834269656, −3.18012459969823296968349578512, −2.05188814638214753610191041462, 0.70528542293496420345448771132, 2.90091462020714430126690557653, 3.81429332898554409211675014827, 5.38996464889462365729108582911, 6.04800720993374050932465652675, 7.46834332425262929018324630645, 8.106613333000413007642276749409, 9.170146470494094362513778209799, 10.16782107604686935506194657565, 11.06267973651422405629713947689

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