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Properties

Label 2-208-16.13-c1-0-14
Degree $2$
Conductor $208$
Sign $0.792 + 0.610i$
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  + (−1.33 + 0.453i)2-s + (0.474 + 0.474i)3-s + (1.58 − 1.21i)4-s + (0.481 − 0.481i)5-s + (−0.851 − 0.420i)6-s − 4.68i·7-s + (−1.57 + 2.34i)8-s − 2.54i·9-s + (−0.426 + 0.864i)10-s + (−1.38 + 1.38i)11-s + (1.33 + 0.176i)12-s + (−0.707 − 0.707i)13-s + (2.12 + 6.26i)14-s + 0.457·15-s + (1.04 − 3.86i)16-s + 4.82·17-s + ⋯
L(s)  = 1  + (−0.947 + 0.320i)2-s + (0.274 + 0.274i)3-s + (0.793 − 0.607i)4-s + (0.215 − 0.215i)5-s + (−0.347 − 0.171i)6-s − 1.76i·7-s + (−0.556 + 0.830i)8-s − 0.849i·9-s + (−0.134 + 0.273i)10-s + (−0.417 + 0.417i)11-s + (0.384 + 0.0509i)12-s + (−0.196 − 0.196i)13-s + (0.567 + 1.67i)14-s + 0.118·15-s + (0.260 − 0.965i)16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.792 + 0.610i$
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: \((2,\ 208,\ (\ :1/2),\ 0.792 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821254 - 0.279663i\)
\(L(\frac12)\) \(\approx\) \(0.821254 - 0.279663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.453i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.474 - 0.474i)T + 3iT^{2} \)
5 \( 1 + (-0.481 + 0.481i)T - 5iT^{2} \)
7 \( 1 + 4.68iT - 7T^{2} \)
11 \( 1 + (1.38 - 1.38i)T - 11iT^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 + (0.235 + 0.235i)T + 19iT^{2} \)
23 \( 1 + 3.34iT - 23T^{2} \)
29 \( 1 + (-5.42 - 5.42i)T + 29iT^{2} \)
31 \( 1 - 0.517T + 31T^{2} \)
37 \( 1 + (-4.08 + 4.08i)T - 37iT^{2} \)
41 \( 1 - 4.40iT - 41T^{2} \)
43 \( 1 + (6.40 - 6.40i)T - 43iT^{2} \)
47 \( 1 + 9.55T + 47T^{2} \)
53 \( 1 + (-3.86 + 3.86i)T - 53iT^{2} \)
59 \( 1 + (5.76 - 5.76i)T - 59iT^{2} \)
61 \( 1 + (-6.84 - 6.84i)T + 61iT^{2} \)
67 \( 1 + (0.991 + 0.991i)T + 67iT^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + 9.98iT - 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 + (-4.07 - 4.07i)T + 83iT^{2} \)
89 \( 1 - 5.90iT - 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−12.17662991783686070046452833908, −10.88825899920314221664789021299, −10.10199507927877694638373944013, −9.535640105155446215252736034231, −8.236653157745802831312877398336, −7.35662436387313732823384230569, −6.44347315707122666043727031294, −4.86259801877877442458201248911, −3.29339925587347295900684249761, −1.05434812168533732400325305686, 2.07697844213521062197529790296, 2.98446484516149416286445576868, 5.33692560569480202130360235867, 6.43908341748149729810880227656, 7.933833436316972640835942747496, 8.397659175875378891029144305796, 9.538571584169531306609132920523, 10.36943864769420337289147711920, 11.60161372426603618401778743279, 12.18005162835986612312197602815

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