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Properties

Label 2-208-13.12-c11-0-69
Degree $2$
Conductor $208$
Sign $-0.845 + 0.534i$
Analytic cond. $159.815$
Root an. cond. $12.6418$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 584.·3-s − 6.97e3i·5-s + 4.11e4i·7-s + 1.64e5·9-s + 5.02e5i·11-s + (1.13e6 − 7.14e5i)13-s − 4.07e6i·15-s − 7.46e6·17-s − 9.86e6i·19-s + 2.40e7i·21-s − 3.09e7·23-s + 1.75e5·25-s − 7.26e6·27-s − 1.95e8·29-s − 1.15e7i·31-s + ⋯
L(s)  = 1  + 1.38·3-s − 0.998i·5-s + 0.925i·7-s + 0.929·9-s + 0.940i·11-s + (0.845 − 0.534i)13-s − 1.38i·15-s − 1.27·17-s − 0.914i·19-s + 1.28i·21-s − 1.00·23-s + 0.00359·25-s − 0.0974·27-s − 1.76·29-s − 0.0722i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(159.815\)
Root analytic conductor: \(12.6418\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :11/2),\ -0.845 + 0.534i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.421284416\)
\(L(\frac12)\) \(\approx\) \(1.421284416\)
\(L(\frac{13}{2})\) 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 \( 1 + (-1.13e6 + 7.14e5i)T \)
good3 \( 1 - 584.T + 1.77e5T^{2} \)
5 \( 1 + 6.97e3iT - 4.88e7T^{2} \)
7 \( 1 - 4.11e4iT - 1.97e9T^{2} \)
11 \( 1 - 5.02e5iT - 2.85e11T^{2} \)
17 \( 1 + 7.46e6T + 3.42e13T^{2} \)
19 \( 1 + 9.86e6iT - 1.16e14T^{2} \)
23 \( 1 + 3.09e7T + 9.52e14T^{2} \)
29 \( 1 + 1.95e8T + 1.22e16T^{2} \)
31 \( 1 + 1.15e7iT - 2.54e16T^{2} \)
37 \( 1 + 1.48e8iT - 1.77e17T^{2} \)
41 \( 1 + 9.79e8iT - 5.50e17T^{2} \)
43 \( 1 - 6.27e6T + 9.29e17T^{2} \)
47 \( 1 - 3.75e8iT - 2.47e18T^{2} \)
53 \( 1 + 9.10e8T + 9.26e18T^{2} \)
59 \( 1 + 1.75e9iT - 3.01e19T^{2} \)
61 \( 1 + 8.61e8T + 4.35e19T^{2} \)
67 \( 1 + 1.06e10iT - 1.22e20T^{2} \)
71 \( 1 + 3.89e9iT - 2.31e20T^{2} \)
73 \( 1 + 6.08e9iT - 3.13e20T^{2} \)
79 \( 1 + 3.82e9T + 7.47e20T^{2} \)
83 \( 1 + 5.34e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.76e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.11e11iT - 7.15e21T^{2} \)
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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

−9.452975575821380343432021269362, −8.987462929564945987516681791534, −8.340875984761824931629136256266, −7.30793134057803230808941889265, −5.84682553925035004858226803563, −4.68997249153083786362394513438, −3.66764712266418870071399925193, −2.37567929099915709413711837588, −1.76758757614183296191731692749, −0.18925308307208090095735925584, 1.43701221252521954307457415373, 2.49560702995211645017569600378, 3.54966613720152588402035131271, 4.04717531332064198766435443973, 6.03625846995318202304424418991, 6.98282686985150653068774123633, 7.945985699457336142356415718455, 8.725513823292956431744684925569, 9.752115104879698253093315204143, 10.77959957152355804857563650830

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