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Properties

Label 2-208-52.23-c0-0-0
Degree $2$
Conductor $208$
Sign $0.859 + 0.511i$
Analytic cond. $0.103805$
Root an. cond. $0.322188$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s − 1.99·25-s + (−0.5 − 0.866i)29-s + (−1.5 + 0.866i)37-s + (1.5 − 0.866i)41-s + (1.49 + 0.866i)45-s + (0.5 + 0.866i)49-s − 53-s + (0.5 − 0.866i)61-s + (1.49 − 0.866i)65-s − 1.73i·73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  − 1.73i·5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s − 1.99·25-s + (−0.5 − 0.866i)29-s + (−1.5 + 0.866i)37-s + (1.5 − 0.866i)41-s + (1.49 + 0.866i)45-s + (0.5 + 0.866i)49-s − 53-s + (0.5 − 0.866i)61-s + (1.49 − 0.866i)65-s − 1.73i·73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.103805\)
Root analytic conductor: \(0.322188\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :0),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6888869348\)
\(L(\frac12)\) \(\approx\) \(0.6888869348\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + 1.73iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−12.60393920561146789535734899655, −11.68685972121358948869191361955, −10.68980671934087732052072348512, −9.310941861310981309639584571100, −8.643663944488015751305887033254, −7.80720753815290391350990065629, −6.13426096303159739593014346481, −5.05956358824527363159912660992, −4.09371039513535875014140741300, −1.83550410882520121517010296642, 2.73420764988430395564816193588, 3.65631048088994838868633934788, 5.62967791504167381176092017247, 6.63249688970102400403475350525, 7.44019572724384792121082795998, 8.802294370848369744271889143508, 9.929997555811770108372414413849, 10.89785009441896763556624464775, 11.46495287541782733866618194983, 12.66946801926131138253777258696

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