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Properties

Label 2-208-13.4-c3-0-2
Degree $2$
Conductor $208$
Sign $-0.726 - 0.687i$
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  + (0.643 − 1.11i)3-s + 15.8i·5-s + (1.02 − 0.592i)7-s + (12.6 + 21.9i)9-s + (−33.6 − 19.4i)11-s + (−45.4 − 11.4i)13-s + (17.7 + 10.2i)15-s + (−18.4 − 31.9i)17-s + (−37.1 + 21.4i)19-s − 1.52i·21-s + (−101. + 176. i)23-s − 127.·25-s + 67.3·27-s + (−29.4 + 51.0i)29-s + 77.3i·31-s + ⋯
L(s)  = 1  + (0.123 − 0.214i)3-s + 1.42i·5-s + (0.0553 − 0.0319i)7-s + (0.469 + 0.812i)9-s + (−0.923 − 0.533i)11-s + (−0.969 − 0.244i)13-s + (0.304 + 0.176i)15-s + (−0.263 − 0.456i)17-s + (−0.449 + 0.259i)19-s − 0.0158i·21-s + (−0.924 + 1.60i)23-s − 1.02·25-s + 0.480·27-s + (−0.188 + 0.327i)29-s + 0.448i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.726 - 0.687i$
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: \((2,\ 208,\ (\ :3/2),\ -0.726 - 0.687i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.380829 + 0.956459i\)
\(L(\frac12)\) \(\approx\) \(0.380829 + 0.956459i\)
\(L(\frac{5}{2})\) 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 \( 1 + (45.4 + 11.4i)T \)
good3 \( 1 + (-0.643 + 1.11i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 15.8iT - 125T^{2} \)
7 \( 1 + (-1.02 + 0.592i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (33.6 + 19.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (18.4 + 31.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (37.1 - 21.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (101. - 176. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (29.4 - 51.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 77.3iT - 2.97e4T^{2} \)
37 \( 1 + (-223. - 129. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-146. - 84.8i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (183. + 317. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 249. iT - 1.03e5T^{2} \)
53 \( 1 - 157.T + 1.48e5T^{2} \)
59 \( 1 + (582. - 336. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (290. + 502. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (156. + 90.4i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-449. + 259. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 982. iT - 3.89e5T^{2} \)
79 \( 1 - 1.26e3T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + (-1.28e3 - 742. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (303. - 175. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.29217545156577201745354020390, −11.08236791597311605596171637669, −10.51499995556124237961376447262, −9.590329515756049511947577288014, −7.86419293474078500744551335732, −7.45771061953158803883863211210, −6.22124813389607206604039870877, −4.92828991440122013700299973752, −3.23297473515889214180028689634, −2.17588029323835282059604188132, 0.40509918730244166318724374059, 2.19960484714657884427254322398, 4.22140234371288922042313135056, 4.88682441038509132995073802393, 6.29742823895375944684610191979, 7.69277595307136254928610558443, 8.657835451462443787739434246722, 9.555861728708257875118868426743, 10.35339059844462716524070802455, 11.84575682765136671345750160043

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