Properties

Label 2-208-13.10-c3-0-4
Degree $2$
Conductor $208$
Sign $-0.957 - 0.289i$
Analytic cond. $12.2723$
Root an. cond. $3.50319$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.41 + 5.91i)3-s + 17.8i·5-s + (−10.6 − 6.16i)7-s + (−9.84 + 17.0i)9-s + (−30.3 + 17.4i)11-s + (32.5 + 33.7i)13-s + (−105. + 60.8i)15-s + (31.2 − 54.1i)17-s + (−21.6 − 12.4i)19-s − 84.2i·21-s + (−83.1 − 144. i)23-s − 192.·25-s + 49.9·27-s + (140. + 243. i)29-s − 18.6i·31-s + ⋯
L(s)  = 1  + (0.657 + 1.13i)3-s + 1.59i·5-s + (−0.576 − 0.332i)7-s + (−0.364 + 0.631i)9-s + (−0.830 + 0.479i)11-s + (0.693 + 0.720i)13-s + (−1.81 + 1.04i)15-s + (0.446 − 0.772i)17-s + (−0.261 − 0.150i)19-s − 0.875i·21-s + (−0.754 − 1.30i)23-s − 1.53·25-s + 0.355·27-s + (0.899 + 1.55i)29-s − 0.108i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ -0.957 - 0.289i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.242748 + 1.64208i\)
\(L(\frac12)\) \(\approx\) \(0.242748 + 1.64208i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-32.5 - 33.7i)T \)
good3 \( 1 + (-3.41 - 5.91i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 17.8iT - 125T^{2} \)
7 \( 1 + (10.6 + 6.16i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (30.3 - 17.4i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-31.2 + 54.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21.6 + 12.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (83.1 + 144. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-140. - 243. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 18.6iT - 2.97e4T^{2} \)
37 \( 1 + (195. - 112. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-53.4 + 30.8i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (69.1 - 119. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 573. iT - 1.03e5T^{2} \)
53 \( 1 - 361.T + 1.48e5T^{2} \)
59 \( 1 + (173. + 100. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (422. - 730. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (267. - 154. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-998. - 576. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 207. iT - 3.89e5T^{2} \)
79 \( 1 - 967.T + 4.93e5T^{2} \)
83 \( 1 + 33.2iT - 5.71e5T^{2} \)
89 \( 1 + (-1.07e3 + 621. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-268. - 154. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.31449568371188746912388362999, −10.92451278913231301984976763172, −10.37712990133521944553756238275, −9.714110958810865870205428174758, −8.556207087323602451421537127288, −7.21219845131238295172680191038, −6.37657950167241373679164615009, −4.63184293226143276276758196716, −3.45205463067572532735189950264, −2.65367160999790257905680690527, 0.64404811709665503802271066236, 1.96574781643569937136313012878, 3.56458054721305500464635606499, 5.30129605274345579220271974948, 6.21801741495154475873383385363, 7.940041662721341695157796521527, 8.187433132766119790495346086710, 9.203543713492268407334938536504, 10.40589642845751238276142884271, 12.04157016749322121243637672244

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