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Properties

Label 2-208-13.3-c7-0-14
Degree $2$
Conductor $208$
Sign $-0.471 - 0.881i$
Analytic cond. $64.9760$
Root an. cond. $8.06077$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.9 + 27.6i)3-s − 54.4·5-s + (−556. + 963. i)7-s + (585. − 1.01e3i)9-s + (3.56e3 + 6.17e3i)11-s + (7.56e3 + 2.34e3i)13-s + (−867. − 1.50e3i)15-s + (1.02e4 − 1.77e4i)17-s + (1.40e4 − 2.43e4i)19-s − 3.54e4·21-s + (1.69e4 + 2.93e4i)23-s − 7.51e4·25-s + 1.07e5·27-s + (8.77e4 + 1.51e5i)29-s − 1.56e4·31-s + ⋯
L(s)  = 1  + (0.340 + 0.590i)3-s − 0.194·5-s + (−0.613 + 1.06i)7-s + (0.267 − 0.463i)9-s + (0.807 + 1.39i)11-s + (0.955 + 0.296i)13-s + (−0.0663 − 0.114i)15-s + (0.506 − 0.876i)17-s + (0.469 − 0.813i)19-s − 0.835·21-s + (0.290 + 0.503i)23-s − 0.962·25-s + 1.04·27-s + (0.668 + 1.15i)29-s − 0.0945·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(64.9760\)
Root analytic conductor: \(8.06077\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :7/2),\ -0.471 - 0.881i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.247268786\)
\(L(\frac12)\) \(\approx\) \(2.247268786\)
\(L(\frac{9}{2})\) 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 \( 1 + (-7.56e3 - 2.34e3i)T \)
good3 \( 1 + (-15.9 - 27.6i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + 54.4T + 7.81e4T^{2} \)
7 \( 1 + (556. - 963. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-3.56e3 - 6.17e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
17 \( 1 + (-1.02e4 + 1.77e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-1.40e4 + 2.43e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (-1.69e4 - 2.93e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-8.77e4 - 1.51e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + 1.56e4T + 2.75e10T^{2} \)
37 \( 1 + (998. + 1.72e3i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (1.11e5 + 1.93e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (2.68e5 - 4.65e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + 5.42e5T + 5.06e11T^{2} \)
53 \( 1 - 1.85e6T + 1.17e12T^{2} \)
59 \( 1 + (6.65e5 - 1.15e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.43e6 - 2.48e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.41e6 + 2.45e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (8.00e5 - 1.38e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 1.39e6T + 1.10e13T^{2} \)
79 \( 1 - 2.33e6T + 1.92e13T^{2} \)
83 \( 1 + 2.37e6T + 2.71e13T^{2} \)
89 \( 1 + (3.52e6 + 6.10e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (4.25e6 - 7.37e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−11.69287593306671377084663954763, −10.17238598708552694689047551257, −9.297142821605990467825386387463, −8.934594298719970194408615138961, −7.30040065752265671448300821105, −6.38146268155069905173905980654, −5.02680450577054515499015346934, −3.87322850338985657369753701000, −2.85314038969573327238300437575, −1.31043157954465461931493570673, 0.58235615849731165194060371079, 1.51098374215917789527966705382, 3.29341464611578854888815881047, 4.01862030606725555017088645683, 5.86511987164250954573639045910, 6.70556248336846947539670702942, 7.901621637259656658246224133943, 8.482957551806704341292879302050, 9.955320639354248175210098142638, 10.73115309618100412777918850967

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