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Properties

Label 2-208-13.10-c3-0-15
Degree $2$
Conductor $208$
Sign $0.275 + 0.961i$
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  + (0.884 + 1.53i)3-s + 0.308i·5-s + (−11.0 − 6.40i)7-s + (11.9 − 20.6i)9-s + (−19.0 + 11.0i)11-s + (10.7 − 45.6i)13-s + (−0.472 + 0.272i)15-s + (35.0 − 60.6i)17-s + (−17.4 − 10.0i)19-s − 22.6i·21-s + (−15.5 − 26.9i)23-s + 124.·25-s + 90.0·27-s + (−42.8 − 74.2i)29-s − 167. i·31-s + ⋯
L(s)  = 1  + (0.170 + 0.294i)3-s + 0.0275i·5-s + (−0.598 − 0.345i)7-s + (0.441 − 0.765i)9-s + (−0.523 + 0.302i)11-s + (0.230 − 0.973i)13-s + (−0.00813 + 0.00469i)15-s + (0.499 − 0.865i)17-s + (−0.210 − 0.121i)19-s − 0.235i·21-s + (−0.141 − 0.244i)23-s + 0.999·25-s + 0.641·27-s + (−0.274 − 0.475i)29-s − 0.970i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.275 + 0.961i$
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: \((2,\ 208,\ (\ :3/2),\ 0.275 + 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.16268 - 0.876696i\)
\(L(\frac12)\) \(\approx\) \(1.16268 - 0.876696i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.7 + 45.6i)T \)
good3 \( 1 + (-0.884 - 1.53i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 0.308iT - 125T^{2} \)
7 \( 1 + (11.0 + 6.40i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (19.0 - 11.0i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-35.0 + 60.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (17.4 + 10.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (15.5 + 26.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (42.8 + 74.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 167. iT - 2.97e4T^{2} \)
37 \( 1 + (19.3 - 11.1i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (124. - 71.5i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-170. + 295. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 224. iT - 1.03e5T^{2} \)
53 \( 1 - 2.40T + 1.48e5T^{2} \)
59 \( 1 + (-353. - 204. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (231. - 401. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (169. - 98.0i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (530. + 306. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 575. iT - 3.89e5T^{2} \)
79 \( 1 + 272.T + 4.93e5T^{2} \)
83 \( 1 - 1.41e3iT - 5.71e5T^{2} \)
89 \( 1 + (-358. + 206. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-726. - 419. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.82096462283514870109697844629, −10.48060947347707304880592846449, −9.907900515682353966350761981529, −8.882046818923676580100277881319, −7.62731666586929222412797219054, −6.64984117563611412802053503616, −5.35668583558868957082060432938, −3.97854571195461997834139747462, −2.83128700385523812894390073893, −0.62394856001691124819339962963, 1.65670548445266015177737162806, 3.14723003070513126025381634482, 4.66271013940366525475772656886, 5.97943356020597758727298777328, 7.05294268517676808881662290795, 8.149573253126277587283018883604, 9.091244455647173015514361661270, 10.25541665542618911846805634825, 11.07000433743649621706103465841, 12.38843724877370894369062223879

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