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Properties

Label 2-208-16.13-c1-0-17
Degree $2$
Conductor $208$
Sign $0.778 + 0.628i$
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  + (1.26 + 0.633i)2-s + (−1.84 − 1.84i)3-s + (1.19 + 1.60i)4-s + (0.917 − 0.917i)5-s + (−1.16 − 3.49i)6-s − 2.65i·7-s + (0.500 + 2.78i)8-s + 3.78i·9-s + (1.74 − 0.579i)10-s + (4.00 − 4.00i)11-s + (0.743 − 5.15i)12-s + (0.707 + 0.707i)13-s + (1.67 − 3.35i)14-s − 3.38·15-s + (−1.13 + 3.83i)16-s − 4.47·17-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)2-s + (−1.06 − 1.06i)3-s + (0.598 + 0.800i)4-s + (0.410 − 0.410i)5-s + (−0.474 − 1.42i)6-s − 1.00i·7-s + (0.176 + 0.984i)8-s + 1.26i·9-s + (0.550 − 0.183i)10-s + (1.20 − 1.20i)11-s + (0.214 − 1.48i)12-s + (0.196 + 0.196i)13-s + (0.448 − 0.895i)14-s − 0.872·15-s + (−0.282 + 0.959i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.778 + 0.628i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.778 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46327 - 0.516978i\)
\(L(\frac12)\) \(\approx\) \(1.46327 - 0.516978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.633i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1.84 + 1.84i)T + 3iT^{2} \)
5 \( 1 + (-0.917 + 0.917i)T - 5iT^{2} \)
7 \( 1 + 2.65iT - 7T^{2} \)
11 \( 1 + (-4.00 + 4.00i)T - 11iT^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + (-0.0709 - 0.0709i)T + 19iT^{2} \)
23 \( 1 - 4.88iT - 23T^{2} \)
29 \( 1 + (-4.01 - 4.01i)T + 29iT^{2} \)
31 \( 1 + 3.70T + 31T^{2} \)
37 \( 1 + (-7.16 + 7.16i)T - 37iT^{2} \)
41 \( 1 - 5.68iT - 41T^{2} \)
43 \( 1 + (7.21 - 7.21i)T - 43iT^{2} \)
47 \( 1 + 4.79T + 47T^{2} \)
53 \( 1 + (-1.67 + 1.67i)T - 53iT^{2} \)
59 \( 1 + (-4.64 + 4.64i)T - 59iT^{2} \)
61 \( 1 + (4.25 + 4.25i)T + 61iT^{2} \)
67 \( 1 + (4.79 + 4.79i)T + 67iT^{2} \)
71 \( 1 - 1.02iT - 71T^{2} \)
73 \( 1 + 4.54iT - 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (-9.06 - 9.06i)T + 83iT^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 - 3.70T + 97T^{2} \)
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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

−12.47504756083736668802272283222, −11.30131665938964494415553881042, −11.14054536427937298978110232774, −9.144234849825266739451930036776, −7.81142878389426002798312056991, −6.72284381227716959349114961120, −6.23281740043888896681608006290, −5.10318425817330119239140760509, −3.71943689743612194259172531846, −1.39346262434038412262891544657, 2.32467965261695986214017280028, 4.11960662079486679907436180965, 4.90066583347728268464232906137, 6.08272622931470525909727487992, 6.67228712144922214229411178508, 9.018441055864043712803467697037, 9.982983654714704596228370164882, 10.61565971383873165472958373041, 11.77407373943406406064912801289, 12.04502516878847684393473515199

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