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Properties

Label 2-208-13.4-c1-0-3
Degree $2$
Conductor $208$
Sign $0.907 + 0.419i$
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  + (−0.693 + 1.20i)3-s − 3.44i·5-s + (2.74 − 1.58i)7-s + (0.538 + 0.933i)9-s + (−2.74 − 1.58i)11-s + (3.59 − 0.278i)13-s + (4.13 + 2.38i)15-s + (0.886 + 1.53i)17-s + (5.54 − 3.20i)19-s + 4.40i·21-s + (−0.693 + 1.20i)23-s − 6.85·25-s − 5.65·27-s + (−2.55 + 4.42i)29-s + 1.35i·31-s + ⋯
L(s)  = 1  + (−0.400 + 0.693i)3-s − 1.53i·5-s + (1.03 − 0.599i)7-s + (0.179 + 0.311i)9-s + (−0.828 − 0.478i)11-s + (0.997 − 0.0773i)13-s + (1.06 + 0.616i)15-s + (0.215 + 0.372i)17-s + (1.27 − 0.734i)19-s + 0.960i·21-s + (−0.144 + 0.250i)23-s − 1.37·25-s − 1.08·27-s + (−0.474 + 0.822i)29-s + 0.242i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.907 + 0.419i$
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: \((2,\ 208,\ (\ :1/2),\ 0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14247 - 0.251514i\)
\(L(\frac12)\) \(\approx\) \(1.14247 - 0.251514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.278i)T \)
good3 \( 1 + (0.693 - 1.20i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.44iT - 5T^{2} \)
7 \( 1 + (-2.74 + 1.58i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.74 + 1.58i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.886 - 1.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.54 + 3.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.693 - 1.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.55 - 4.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.35iT - 31T^{2} \)
37 \( 1 + (9.79 + 5.65i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.96 + 1.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.13 - 3.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.57iT - 47T^{2} \)
53 \( 1 - 3.73T + 53T^{2} \)
59 \( 1 + (11.7 - 6.76i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.40 - 9.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.882 + 0.509i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.5 + 6.66i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.33iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 + (-4.00 - 2.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.2 - 5.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.28828471800032550958435773436, −11.13346657519150146946442965886, −10.60267283807342816171325384010, −9.314471909513340247676994609950, −8.385153110544956654659386832514, −7.53797074734665867802877977888, −5.44910061714024554118331012876, −5.04266290356808980960021120421, −3.88871376748124435551611292281, −1.29407300279499581242131621102, 1.92754059869991195746775356720, 3.44547201609088549855655285979, 5.32833352580661335665128032008, 6.36404869330524468409123126158, 7.33759988664697879406827445688, 8.121861612717755164892551508848, 9.710914221971227441252892929301, 10.71486266539157274399122107342, 11.56177773541766260852315319311, 12.17782600592236128935958968249

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