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Properties

Label 2-208-13.4-c3-0-13
Degree $2$
Conductor $208$
Sign $0.729 + 0.684i$
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 − 1.73i)3-s − 1.73i·5-s + (12 − 6.92i)7-s + (11.5 + 19.9i)9-s + (−12 − 6.92i)11-s + (45.5 + 11.2i)13-s + (−2.99 − 1.73i)15-s + (−58.5 − 101. i)17-s + (99 − 57.1i)19-s − 27.7i·21-s + (−39 + 67.5i)23-s + 122·25-s + 100·27-s + (70.5 − 122. i)29-s − 155. i·31-s + ⋯
L(s)  = 1  + (0.192 − 0.333i)3-s − 0.154i·5-s + (0.647 − 0.374i)7-s + (0.425 + 0.737i)9-s + (−0.328 − 0.189i)11-s + (0.970 + 0.240i)13-s + (−0.0516 − 0.0298i)15-s + (−0.834 − 1.44i)17-s + (1.19 − 0.690i)19-s − 0.287i·21-s + (−0.353 + 0.612i)23-s + 0.975·25-s + 0.712·27-s + (0.451 − 0.781i)29-s − 0.903i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.729 + 0.684i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ 0.729 + 0.684i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.93660 - 0.766034i\)
\(L(\frac12)\) \(\approx\) \(1.93660 - 0.766034i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-45.5 - 11.2i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 1.73iT - 125T^{2} \)
7 \( 1 + (-12 + 6.92i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (12 + 6.92i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (58.5 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-99 + 57.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (39 - 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.5 + 122. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 155. iT - 2.97e4T^{2} \)
37 \( 1 + (124.5 + 71.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-235.5 - 135. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-52 - 90.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 301. iT - 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (-246 + 142. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.5 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-681 - 393. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (915 - 528. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 458. iT - 3.89e5T^{2} \)
79 \( 1 + 1.27e3T + 4.93e5T^{2} \)
83 \( 1 - 789. iT - 5.71e5T^{2} \)
89 \( 1 + (846 + 488. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-174 + 100. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.56885243740790785001834536092, −11.09267344104038232102105219269, −9.835781014522031616098884338933, −8.727149893366397299792231971338, −7.72567831698769910932769734207, −6.92495851557559707457824789907, −5.35594562269790954217406331365, −4.34095374656694777953470558090, −2.60138688882169316357273968957, −1.04631267028510669576583039583, 1.45029455959214390114118125828, 3.24767138576359516999663601721, 4.43817892699905125639082933181, 5.77872364940037632261564910583, 6.89956670413125727925317936518, 8.283761099719966662700360282901, 8.934129358721195011190726339742, 10.25109597878612378579809896786, 10.89733847083912375357617703906, 12.14674524305745409670587747741

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