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Properties

Label 2-26e2-13.12-c1-0-11
Degree $2$
Conductor $676$
Sign $0.832 + 0.554i$
Analytic cond. $5.39788$
Root an. cond. $2.32333$
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  + 3·3-s − 2i·5-s + i·7-s + 6·9-s − 5i·11-s − 6i·15-s − 3·17-s + 3i·19-s + 3i·21-s + 23-s + 25-s + 9·27-s − 29-s + 8i·31-s − 15i·33-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894i·5-s + 0.377i·7-s + 2·9-s − 1.50i·11-s − 1.54i·15-s − 0.727·17-s + 0.688i·19-s + 0.654i·21-s + 0.208·23-s + 0.200·25-s + 1.73·27-s − 0.185·29-s + 1.43i·31-s − 2.61i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53474 - 0.767458i\)
\(L(\frac12)\) \(\approx\) \(2.53474 - 0.767458i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 3T + 3T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 3iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 + 3iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 5iT - 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 11iT - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−10.17202881622040707764955737291, −9.127544497112870507264193080840, −8.662746854602289561990604808525, −8.261328396577919443065384834842, −7.15694249095802160384049077075, −5.89840135772017185002172497898, −4.70452276948966902557912846616, −3.60228677617363761986581441852, −2.75327122845791427970053447273, −1.39648513513354279490734332798, 1.99478164153832457017927903455, 2.75028703069190537573406129864, 3.85425013509095682508339906848, 4.69951632223728736102174083352, 6.57348752635174227179517644059, 7.27339282853041956826070300138, 7.83459523416562121572754147898, 8.974525682918489055786000328204, 9.580270842061132014822750683554, 10.34154796304074901253268792288

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