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Properties

Label 2-208-13.12-c3-0-13
Degree $2$
Conductor $208$
Sign $0.786 + 0.618i$
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  + 7.78·3-s − 7.31i·5-s − 0.135i·7-s + 33.6·9-s − 48.8i·11-s + (36.8 + 28.9i)13-s − 56.9i·15-s + 68.4·17-s − 83.8i·19-s − 1.05i·21-s − 11.7·23-s + 71.5·25-s + 51.5·27-s − 177.·29-s + 197. i·31-s + ⋯
L(s)  = 1  + 1.49·3-s − 0.654i·5-s − 0.00733i·7-s + 1.24·9-s − 1.33i·11-s + (0.786 + 0.618i)13-s − 0.980i·15-s + 0.976·17-s − 1.01i·19-s − 0.0109i·21-s − 0.106·23-s + 0.572·25-s + 0.367·27-s − 1.13·29-s + 1.14i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.786 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.786 + 0.618i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ 0.786 + 0.618i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.78381 - 0.963522i\)
\(L(\frac12)\) \(\approx\) \(2.78381 - 0.963522i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-36.8 - 28.9i)T \)
good3 \( 1 - 7.78T + 27T^{2} \)
5 \( 1 + 7.31iT - 125T^{2} \)
7 \( 1 + 0.135iT - 343T^{2} \)
11 \( 1 + 48.8iT - 1.33e3T^{2} \)
17 \( 1 - 68.4T + 4.91e3T^{2} \)
19 \( 1 + 83.8iT - 6.85e3T^{2} \)
23 \( 1 + 11.7T + 1.21e4T^{2} \)
29 \( 1 + 177.T + 2.43e4T^{2} \)
31 \( 1 - 197. iT - 2.97e4T^{2} \)
37 \( 1 - 283. iT - 5.06e4T^{2} \)
41 \( 1 - 70.5iT - 6.89e4T^{2} \)
43 \( 1 + 28.3T + 7.95e4T^{2} \)
47 \( 1 + 536. iT - 1.03e5T^{2} \)
53 \( 1 + 5.84T + 1.48e5T^{2} \)
59 \( 1 - 304. iT - 2.05e5T^{2} \)
61 \( 1 + 378.T + 2.26e5T^{2} \)
67 \( 1 + 698. iT - 3.00e5T^{2} \)
71 \( 1 - 922. iT - 3.57e5T^{2} \)
73 \( 1 - 1.14e3iT - 3.89e5T^{2} \)
79 \( 1 + 464.T + 4.93e5T^{2} \)
83 \( 1 - 328. iT - 5.71e5T^{2} \)
89 \( 1 - 737. iT - 7.04e5T^{2} \)
97 \( 1 - 944. iT - 9.12e5T^{2} \)
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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

−11.91395641540654996162508673326, −10.80271869033109298401689937199, −9.481995540442140006235579334090, −8.699136013667255732435324289619, −8.243079764424511171572803660468, −6.90748905721658667138928882484, −5.41139002559727865008750813502, −3.88831474503384621114713173810, −2.92041289370989314371867382503, −1.24465950771341121895850327436, 1.84011882291045086058812304609, 3.08588365442825641071654180582, 4.06535276006804149877221962784, 5.85796009056497735537697064624, 7.39485944982838112723845009681, 7.86984544884508135881194139039, 9.111723342747853581912582328875, 9.911083790403727825527635023971, 10.82632552717799645215809916577, 12.29659576755815838976748490557

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