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Properties

Label 2-208-13.12-c9-0-46
Degree $2$
Conductor $208$
Sign $0.965 + 0.259i$
Analytic cond. $107.127$
Root an. cond. $10.3502$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 241.·3-s − 9.99e2i·5-s + 3.39e3i·7-s + 3.84e4·9-s + 5.45e4i·11-s + (−9.94e4 − 2.67e4i)13-s − 2.40e5i·15-s + 6.43e5·17-s − 1.06e6i·19-s + 8.17e5i·21-s − 7.81e5·23-s + 9.53e5·25-s + 4.51e6·27-s + 1.44e6·29-s − 4.53e5i·31-s + ⋯
L(s)  = 1  + 1.71·3-s − 0.715i·5-s + 0.534i·7-s + 1.95·9-s + 1.12i·11-s + (−0.965 − 0.259i)13-s − 1.22i·15-s + 1.86·17-s − 1.87i·19-s + 0.917i·21-s − 0.582·23-s + 0.488·25-s + 1.63·27-s + 0.380·29-s − 0.0881i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(107.127\)
Root analytic conductor: \(10.3502\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :9/2),\ 0.965 + 0.259i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.734274281\)
\(L(\frac12)\) \(\approx\) \(4.734274281\)
\(L(\frac{11}{2})\) 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 \( 1 + (9.94e4 + 2.67e4i)T \)
good3 \( 1 - 241.T + 1.96e4T^{2} \)
5 \( 1 + 9.99e2iT - 1.95e6T^{2} \)
7 \( 1 - 3.39e3iT - 4.03e7T^{2} \)
11 \( 1 - 5.45e4iT - 2.35e9T^{2} \)
17 \( 1 - 6.43e5T + 1.18e11T^{2} \)
19 \( 1 + 1.06e6iT - 3.22e11T^{2} \)
23 \( 1 + 7.81e5T + 1.80e12T^{2} \)
29 \( 1 - 1.44e6T + 1.45e13T^{2} \)
31 \( 1 + 4.53e5iT - 2.64e13T^{2} \)
37 \( 1 - 1.33e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.91e7iT - 3.27e14T^{2} \)
43 \( 1 - 1.95e7T + 5.02e14T^{2} \)
47 \( 1 - 5.93e7iT - 1.11e15T^{2} \)
53 \( 1 - 9.92e7T + 3.29e15T^{2} \)
59 \( 1 - 3.96e7iT - 8.66e15T^{2} \)
61 \( 1 - 9.22e7T + 1.16e16T^{2} \)
67 \( 1 + 1.69e8iT - 2.72e16T^{2} \)
71 \( 1 - 7.66e7iT - 4.58e16T^{2} \)
73 \( 1 + 2.76e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.12e8T + 1.19e17T^{2} \)
83 \( 1 - 1.34e8iT - 1.86e17T^{2} \)
89 \( 1 - 5.95e8iT - 3.50e17T^{2} \)
97 \( 1 - 3.79e8iT - 7.60e17T^{2} \)
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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

−10.23569159429454192160705649691, −9.462601957772251705087531176722, −8.830674346591507470999433405395, −7.81823490624669278028640713153, −7.12822034793230643730735037662, −5.24750802055937656415352543359, −4.31404222424349569258096475242, −2.93885195573755768093964777635, −2.23574402444482180347818419421, −0.949270361168360221698095869957, 1.04478949246074186247985797223, 2.33048001460765526703333765711, 3.31374075335013537474874143722, 3.91380395941601161987531761324, 5.71528221894488532768394424843, 7.17531696358159505467219067763, 7.84562843284642505569753446460, 8.643010540436251153060232304683, 9.970648997728384216369092191073, 10.25929442571455213637295523395

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