Properties

Label 2-208-13.9-c3-0-3
Degree $2$
Conductor $208$
Sign $-0.477 - 0.878i$
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  + (−1.5 + 2.59i)3-s + 2·5-s + (−2.5 − 4.33i)7-s + (9 + 15.5i)9-s + (6.5 − 11.2i)11-s + (−13 + 45.0i)13-s + (−3 + 5.19i)15-s + (−13.5 − 23.3i)17-s + (37.5 + 64.9i)19-s + 15.0·21-s + (−93.5 + 161. i)23-s − 121·25-s − 135·27-s + (6.5 − 11.2i)29-s + 104·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 0.178·5-s + (−0.134 − 0.233i)7-s + (0.333 + 0.577i)9-s + (0.178 − 0.308i)11-s + (−0.277 + 0.960i)13-s + (−0.0516 + 0.0894i)15-s + (−0.192 − 0.333i)17-s + (0.452 + 0.784i)19-s + 0.155·21-s + (−0.847 + 1.46i)23-s − 0.967·25-s − 0.962·27-s + (0.0416 − 0.0720i)29-s + 0.602·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.608410 + 1.02326i\)
\(L(\frac12)\) \(\approx\) \(0.608410 + 1.02326i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13 - 45.0i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 2T + 125T^{2} \)
7 \( 1 + (2.5 + 4.33i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-6.5 + 11.2i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (13.5 + 23.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-37.5 - 64.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (93.5 - 161. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-6.5 + 11.2i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 104T + 2.97e4T^{2} \)
37 \( 1 + (211.5 - 366. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (97.5 - 168. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-99.5 - 172. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 388T + 1.03e5T^{2} \)
53 \( 1 - 618T + 1.48e5T^{2} \)
59 \( 1 + (-245.5 - 425. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (87.5 + 151. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-408.5 + 707. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-39.5 - 68.4i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 230T + 3.89e5T^{2} \)
79 \( 1 + 764T + 4.93e5T^{2} \)
83 \( 1 - 732T + 5.71e5T^{2} \)
89 \( 1 + (-520.5 + 901. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-48.5 - 84.0i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.94867983439193308755260365541, −11.40238212883272602709171364392, −10.06882961087132669315371485411, −9.680525912107763221143584986997, −8.245865498698547668179080522751, −7.16708000909107072722337546958, −5.92367806920310014654912159924, −4.78499932156365963939325457382, −3.64178527322688112461103404606, −1.76304508559741831293556507119, 0.52110203974842657410224456393, 2.26893635833892565985085657895, 3.92349284921629526534641331751, 5.43148244157620446685924649427, 6.46118646899220112430549569257, 7.40375909817818841950814940249, 8.610949367658929806491224771094, 9.718269207752830463541414651598, 10.59265515755958299685065969329, 11.89651639976914559990639936797

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