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Properties

Label 2-208-13.10-c3-0-12
Degree $2$
Conductor $208$
Sign $0.999 + 0.00872i$
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  + (2.97 + 5.15i)3-s − 13.0i·5-s + (3.11 + 1.79i)7-s + (−4.23 + 7.34i)9-s + (24.2 − 14.0i)11-s + (40.6 + 23.2i)13-s + (67.2 − 38.8i)15-s + (57.7 − 100. i)17-s + (−112. − 64.7i)19-s + 21.4i·21-s + (87.4 + 151. i)23-s − 44.8·25-s + 110.·27-s + (109. + 189. i)29-s − 53.3i·31-s + ⋯
L(s)  = 1  + (0.573 + 0.992i)3-s − 1.16i·5-s + (0.168 + 0.0970i)7-s + (−0.156 + 0.271i)9-s + (0.665 − 0.384i)11-s + (0.868 + 0.496i)13-s + (1.15 − 0.668i)15-s + (0.823 − 1.42i)17-s + (−1.35 − 0.781i)19-s + 0.222i·21-s + (0.792 + 1.37i)23-s − 0.358·25-s + 0.786·27-s + (0.701 + 1.21i)29-s − 0.309i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.999 + 0.00872i$
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: \((2,\ 208,\ (\ :3/2),\ 0.999 + 0.00872i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.32075 - 0.0101192i\)
\(L(\frac12)\) \(\approx\) \(2.32075 - 0.0101192i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-40.6 - 23.2i)T \)
good3 \( 1 + (-2.97 - 5.15i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 13.0iT - 125T^{2} \)
7 \( 1 + (-3.11 - 1.79i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.2 + 14.0i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-57.7 + 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (112. + 64.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-87.4 - 151. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-109. - 189. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 53.3iT - 2.97e4T^{2} \)
37 \( 1 + (-109. + 63.3i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (222. - 128. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-14.2 + 24.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 108. iT - 1.03e5T^{2} \)
53 \( 1 - 215.T + 1.48e5T^{2} \)
59 \( 1 + (339. + 196. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-112. + 194. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-120. + 69.7i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (550. + 317. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 946. iT - 3.89e5T^{2} \)
79 \( 1 - 198.T + 4.93e5T^{2} \)
83 \( 1 + 692. iT - 5.71e5T^{2} \)
89 \( 1 + (701. - 405. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (996. + 575. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.85322400075076865885216736295, −10.96912611904634988059124046067, −9.610007643657040692870321494645, −9.029298349945688944879629725705, −8.411114990219180828382986301711, −6.80414608314323423781886805191, −5.25487478054661027734360665902, −4.38498530775257023548882436032, −3.23826770851956605683239991858, −1.14631350925576957342251208737, 1.46724383642385783787308274554, 2.75442323753051429790132655213, 4.07510028556714621829668936014, 6.17871992551889050787767222082, 6.75490785233466142655858498332, 7.955975623314296672652348648607, 8.561311564734548557075330768280, 10.29717252084900095190010139148, 10.75675840222851284378226486757, 12.17423975235155116378295709743

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