Properties

Label 2-208-13.12-c5-0-28
Degree $2$
Conductor $208$
Sign $-0.981 + 0.192i$
Analytic cond. $33.3598$
Root an. cond. $5.77579$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13·3-s + 51i·5-s − 105i·7-s − 74·9-s − 120i·11-s + (−598 + 117i)13-s + 663i·15-s − 1.10e3·17-s + 1.17e3i·19-s − 1.36e3i·21-s − 1.05e3·23-s + 524·25-s − 4.12e3·27-s − 4.10e3·29-s − 9.62e3i·31-s + ⋯
L(s)  = 1  + 0.833·3-s + 0.912i·5-s − 0.809i·7-s − 0.304·9-s − 0.299i·11-s + (−0.981 + 0.192i)13-s + 0.760i·15-s − 0.923·17-s + 0.743i·19-s − 0.675i·21-s − 0.413·23-s + 0.167·25-s − 1.08·27-s − 0.906·29-s − 1.79i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.981 + 0.192i$
Analytic conductor: \(33.3598\)
Root analytic conductor: \(5.77579\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 208,\ (\ :5/2),\ -0.981 + 0.192i)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + (598 - 117i)T \)
good3 \( 1 - 13T + 243T^{2} \)
5 \( 1 - 51iT - 3.12e3T^{2} \)
7 \( 1 + 105iT - 1.68e4T^{2} \)
11 \( 1 + 120iT - 1.61e5T^{2} \)
17 \( 1 + 1.10e3T + 1.41e6T^{2} \)
19 \( 1 - 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.05e3T + 6.43e6T^{2} \)
29 \( 1 + 4.10e3T + 2.05e7T^{2} \)
31 \( 1 + 9.62e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.70e3iT - 6.93e7T^{2} \)
41 \( 1 + 9.48e3iT - 1.15e8T^{2} \)
43 \( 1 + 9.99e3T + 1.47e8T^{2} \)
47 \( 1 - 2.94e3iT - 2.29e8T^{2} \)
53 \( 1 + 750T + 4.18e8T^{2} \)
59 \( 1 - 4.09e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.79e4T + 8.44e8T^{2} \)
67 \( 1 + 2.28e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.37e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.88e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.32e4T + 3.07e9T^{2} \)
83 \( 1 + 5.54e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.04e5iT - 5.58e9T^{2} \)
97 \( 1 - 1.60e5iT - 8.58e9T^{2} \)
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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

−10.95845871089978335347092100929, −10.07240061383184630196794295625, −9.086674368057515208421865010271, −7.903525462519486332437382595860, −7.16193026398581799201107308011, −5.97710239461086149698138786710, −4.28231856942894561607070965770, −3.17317319576574243793865835401, −2.10018479302355757072476808650, 0, 1.91835734953298521873449231735, 2.97835122906630426785308206674, 4.58840901055223369668989596583, 5.52351941941242903267655947756, 7.04041124850421519035628995697, 8.281876119752454822123031392567, 8.935208404243323166050664514314, 9.613437076527661460268641206209, 11.07518505427642931373588017154

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