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Properties

Label 2-855-95.94-c0-0-0
Degree $2$
Conductor $855$
Sign $-i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
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  − 4-s + i·5-s + 16-s + 2i·17-s − 19-s − i·20-s + 2i·23-s − 25-s − 2i·47-s + 49-s + 2·61-s − 64-s − 2i·68-s + 76-s + i·80-s + ⋯
L(s)  = 1  − 4-s + i·5-s + 16-s + 2i·17-s − 19-s − i·20-s + 2i·23-s − 25-s − 2i·47-s + 49-s + 2·61-s − 64-s − 2i·68-s + 76-s + i·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6862839031\)
\(L(\frac12)\) \(\approx\) \(0.6862839031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - iT \)
19 \( 1 + T \)
good2 \( 1 + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 2iT - T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.39362992561294684367144717927, −9.938023508716114438615412539379, −8.844812329386031709516243281533, −8.159786214993360024317458354109, −7.23354116876202208422031411645, −6.16485284536393311714688905889, −5.43089577767191624653108792806, −4.04005117855751908584612703180, −3.51522965472445254476240749136, −1.90073524500712755725850802148, 0.73410684909916266227780964106, 2.59228109815596430842835535724, 4.14212187204367362594602075666, 4.72730467283073990681562354768, 5.52956726641953847845006122904, 6.71051172070228487336639076565, 7.896313259826873230734736719258, 8.622554972135854310651242063969, 9.233680695753324646036289475510, 9.935224915325706876146880644565

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