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Properties

Label 2-208-13.10-c3-0-8
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: $\chi_{208} (49, \cdot )$
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

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

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