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Properties

Label 2-208-13.10-c3-0-3
Degree $2$
Conductor $208$
Sign $0.916 - 0.399i$
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  + (−4.89 − 8.47i)3-s + 2.49i·5-s + (15.5 + 8.98i)7-s + (−34.4 + 59.5i)9-s + (−49.6 + 28.6i)11-s + (46.6 + 4.66i)13-s + (21.1 − 12.2i)15-s + (−8.75 + 15.1i)17-s + (−19.1 − 11.0i)19-s − 175. i·21-s + (65.5 + 113. i)23-s + 118.·25-s + 409.·27-s + (−37.1 − 64.3i)29-s + 110. i·31-s + ⋯
L(s)  = 1  + (−0.941 − 1.63i)3-s + 0.223i·5-s + (0.840 + 0.485i)7-s + (−1.27 + 2.20i)9-s + (−1.35 + 0.784i)11-s + (0.995 + 0.0995i)13-s + (0.364 − 0.210i)15-s + (−0.124 + 0.216i)17-s + (−0.231 − 0.133i)19-s − 1.82i·21-s + (0.594 + 1.03i)23-s + 0.950·25-s + 2.91·27-s + (−0.237 − 0.411i)29-s + 0.639i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.916 - 0.399i$
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: \((2,\ 208,\ (\ :3/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.969359 + 0.202101i\)
\(L(\frac12)\) \(\approx\) \(0.969359 + 0.202101i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.6 - 4.66i)T \)
good3 \( 1 + (4.89 + 8.47i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 2.49iT - 125T^{2} \)
7 \( 1 + (-15.5 - 8.98i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (49.6 - 28.6i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (8.75 - 15.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (19.1 + 11.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-65.5 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (37.1 + 64.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 110. iT - 2.97e4T^{2} \)
37 \( 1 + (-209. + 120. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (369. - 213. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 158. iT - 1.03e5T^{2} \)
53 \( 1 - 47.9T + 1.48e5T^{2} \)
59 \( 1 + (-382. - 220. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (434. - 752. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (693. - 400. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-773. - 446. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 413. iT - 3.89e5T^{2} \)
79 \( 1 - 246.T + 4.93e5T^{2} \)
83 \( 1 + 333. iT - 5.71e5T^{2} \)
89 \( 1 + (328. - 189. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-815. - 470. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.98465838849184016098744046662, −11.23464711454404756691201426041, −10.51015427261501825930905854275, −8.621453740793932562679505629677, −7.74913024623450630988374742111, −6.94400543157742098240811533967, −5.79434547100813138837180328368, −4.97725338034322706761749980549, −2.48039521217116168990388809661, −1.31772445263290893038022461808, 0.52407146000275570176585492377, 3.28662367707533891565821997245, 4.57752261818654201396287562946, 5.23352334084720375823991315829, 6.33009700776400877913444461574, 8.135113872740061657527985387679, 8.998179162757947754111060763039, 10.29348760601169549543256049141, 10.88012636471748963024693194798, 11.33427041711891549151377932405

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