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Properties

Label 2-208-16.5-c1-0-16
Degree $2$
Conductor $208$
Sign $-0.115 + 0.993i$
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  + (−1.40 − 0.179i)2-s + (1.23 − 1.23i)3-s + (1.93 + 0.503i)4-s + (−0.845 − 0.845i)5-s + (−1.94 + 1.50i)6-s − 3.63i·7-s + (−2.62 − 1.05i)8-s − 0.0264i·9-s + (1.03 + 1.33i)10-s + (−0.210 − 0.210i)11-s + (3.00 − 1.76i)12-s + (0.707 − 0.707i)13-s + (−0.651 + 5.09i)14-s − 2.08·15-s + (3.49 + 1.94i)16-s − 1.19·17-s + ⋯
L(s)  = 1  + (−0.991 − 0.126i)2-s + (0.710 − 0.710i)3-s + (0.967 + 0.251i)4-s + (−0.378 − 0.378i)5-s + (−0.794 + 0.614i)6-s − 1.37i·7-s + (−0.928 − 0.372i)8-s − 0.00880i·9-s + (0.327 + 0.423i)10-s + (−0.0635 − 0.0635i)11-s + (0.866 − 0.508i)12-s + (0.196 − 0.196i)13-s + (−0.174 + 1.36i)14-s − 0.537·15-s + (0.873 + 0.487i)16-s − 0.289·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.115 + 0.993i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.115 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594282 - 0.667622i\)
\(L(\frac12)\) \(\approx\) \(0.594282 - 0.667622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.179i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.23 + 1.23i)T - 3iT^{2} \)
5 \( 1 + (0.845 + 0.845i)T + 5iT^{2} \)
7 \( 1 + 3.63iT - 7T^{2} \)
11 \( 1 + (0.210 + 0.210i)T + 11iT^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 + (0.311 - 0.311i)T - 19iT^{2} \)
23 \( 1 + 5.29iT - 23T^{2} \)
29 \( 1 + (1.97 - 1.97i)T - 29iT^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 + (3.79 + 3.79i)T + 37iT^{2} \)
41 \( 1 - 8.66iT - 41T^{2} \)
43 \( 1 + (-8.04 - 8.04i)T + 43iT^{2} \)
47 \( 1 - 1.19T + 47T^{2} \)
53 \( 1 + (-6.02 - 6.02i)T + 53iT^{2} \)
59 \( 1 + (-5.21 - 5.21i)T + 59iT^{2} \)
61 \( 1 + (5.69 - 5.69i)T - 61iT^{2} \)
67 \( 1 + (5.71 - 5.71i)T - 67iT^{2} \)
71 \( 1 + 4.32iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + (-1.50 + 1.50i)T - 83iT^{2} \)
89 \( 1 + 4.33iT - 89T^{2} \)
97 \( 1 - 7.63T + 97T^{2} \)
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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

−12.13129328771022177602599428687, −10.87813904809483444739604996436, −10.24988242696725498882630224821, −8.897896720192375810101919592420, −8.075014146666260804164663875035, −7.43685742209769204270257192831, −6.44480654548695431780311236723, −4.29700031174346760898882453740, −2.72079718019144037867128807195, −1.03392619956931822258106674903, 2.35864391869068277351976784016, 3.56225818586944235546548701338, 5.47276550102903966308361583906, 6.70483078928957652385586765882, 7.971510690945758215292696544658, 8.903208179365003566071583117468, 9.402386097088554136706354104937, 10.46551615553323627288066048813, 11.56414198716662942345299456205, 12.22431425543248725099277109699

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