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Properties

Label 2-208-13.12-c9-0-52
Degree $2$
Conductor $208$
Sign $-0.420 + 0.907i$
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  + 144.·3-s + 1.01e3i·5-s + 5.65e3i·7-s + 1.05e3·9-s − 6.61e4i·11-s + (4.32e4 − 9.34e4i)13-s + 1.46e5i·15-s + 1.00e5·17-s − 1.88e5i·19-s + 8.14e5i·21-s − 2.22e6·23-s + 9.13e5·25-s − 2.68e6·27-s − 5.75e6·29-s + 7.19e6i·31-s + ⋯
L(s)  = 1  + 1.02·3-s + 0.729i·5-s + 0.890i·7-s + 0.0536·9-s − 1.36i·11-s + (0.420 − 0.907i)13-s + 0.748i·15-s + 0.292·17-s − 0.331i·19-s + 0.914i·21-s − 1.65·23-s + 0.467·25-s − 0.971·27-s − 1.51·29-s + 1.39i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.420 + 0.907i$
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: \((2,\ 208,\ (\ :9/2),\ -0.420 + 0.907i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.185135136\)
\(L(\frac12)\) \(\approx\) \(1.185135136\)
\(L(\frac{11}{2})\) 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 \( 1 + (-4.32e4 + 9.34e4i)T \)
good3 \( 1 - 144.T + 1.96e4T^{2} \)
5 \( 1 - 1.01e3iT - 1.95e6T^{2} \)
7 \( 1 - 5.65e3iT - 4.03e7T^{2} \)
11 \( 1 + 6.61e4iT - 2.35e9T^{2} \)
17 \( 1 - 1.00e5T + 1.18e11T^{2} \)
19 \( 1 + 1.88e5iT - 3.22e11T^{2} \)
23 \( 1 + 2.22e6T + 1.80e12T^{2} \)
29 \( 1 + 5.75e6T + 1.45e13T^{2} \)
31 \( 1 - 7.19e6iT - 2.64e13T^{2} \)
37 \( 1 - 3.71e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.77e6iT - 3.27e14T^{2} \)
43 \( 1 - 1.44e7T + 5.02e14T^{2} \)
47 \( 1 + 6.34e7iT - 1.11e15T^{2} \)
53 \( 1 + 8.43e7T + 3.29e15T^{2} \)
59 \( 1 + 1.12e8iT - 8.66e15T^{2} \)
61 \( 1 + 1.70e8T + 1.16e16T^{2} \)
67 \( 1 - 4.94e7iT - 2.72e16T^{2} \)
71 \( 1 + 1.17e8iT - 4.58e16T^{2} \)
73 \( 1 + 2.58e8iT - 5.88e16T^{2} \)
79 \( 1 - 4.68e8T + 1.19e17T^{2} \)
83 \( 1 + 5.71e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.75e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.58e8iT - 7.60e17T^{2} \)
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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

−10.45781385163968971711934516542, −9.196869336986924423577211678362, −8.471303838541647628331089933048, −7.72639907748172348581363815869, −6.25252838152409279364469976171, −5.45509338989076449698135102383, −3.49572695722950101095367076816, −3.04341846645273873342938317143, −1.92322676426101513842337293458, −0.19677738218879466869057866186, 1.36647043369787659637434906426, 2.26974854712807701774281996969, 3.84309053759768653335555189655, 4.42821121534186933870621862812, 5.97956032838626437880634689250, 7.41246741539925645425669350809, 7.992664294396124016000487600828, 9.235276577310623711637639612251, 9.694241982398575457771034367396, 10.99854032512885082992400081346

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