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Properties

Label 2-208-13.4-c1-0-5
Degree $2$
Conductor $208$
Sign $0.252 + 0.967i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)3-s − 1.73i·5-s + (−0.499 − 0.866i)9-s + (−2.5 − 2.59i)13-s + (−2.99 − 1.73i)15-s + (1.5 + 2.59i)17-s + (3 − 1.73i)19-s + (−3 + 5.19i)23-s + 2.00·25-s + 4.00·27-s + (−1.5 + 2.59i)29-s − 3.46i·31-s + (7.5 + 4.33i)37-s + (−7 + 1.73i)39-s + (−4.5 − 2.59i)41-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s − 0.774i·5-s + (−0.166 − 0.288i)9-s + (−0.693 − 0.720i)13-s + (−0.774 − 0.447i)15-s + (0.363 + 0.630i)17-s + (0.688 − 0.397i)19-s + (−0.625 + 1.08i)23-s + 0.400·25-s + 0.769·27-s + (−0.278 + 0.482i)29-s − 0.622i·31-s + (1.23 + 0.711i)37-s + (−1.12 + 0.277i)39-s + (−0.702 − 0.405i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11251 - 0.859343i\)
\(L(\frac12)\) \(\approx\) \(1.11251 - 0.859343i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 2.59i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-7.5 - 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (6 - 3.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.54699818345103306855453505014, −11.45495027759070024819545450185, −10.07646555857860327357034591417, −9.069187890029816210180508184004, −7.959994510590298019258233541229, −7.45106848782534714074151101213, −5.98207289863322978693851937168, −4.73761792566358395716962877283, −2.97268242908760213506829911682, −1.38448464762232577824345710124, 2.62342822046411366280422592912, 3.79154668879043519867940309316, 4.94773251899525172161556006660, 6.50352133434169534190270262464, 7.57484645845205865542897602846, 8.859267307832874707492344657951, 9.763204894646968053002029710742, 10.38562472692168141645440280043, 11.51221617533675661513348033336, 12.47336725164043017386172476863

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