↑

Properties

Label 2-920-920.699-c0-0-1
Degree $2$
Conductor $920$
Sign $0.899 + 0.436i$
Analytic cond. $0.459139$
Root an. cond. $0.677598$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.797 − 1.74i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−1.61 − 0.474i)11-s + (0.118 + 0.258i)13-s + (0.273 − 1.89i)14-s + (0.415 − 0.909i)16-s + (0.142 + 0.989i)18-s + (−1.10 + 0.708i)19-s + (0.654 + 0.755i)20-s − 1.68·22-s + (0.654 + 0.755i)23-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.797 − 1.74i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−1.61 − 0.474i)11-s + (0.118 + 0.258i)13-s + (0.273 − 1.89i)14-s + (0.415 − 0.909i)16-s + (0.142 + 0.989i)18-s + (−1.10 + 0.708i)19-s + (0.654 + 0.755i)20-s − 1.68·22-s + (0.654 + 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ 0.899 + 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.786895147\)
\(L(\frac12)\) \(\approx\) \(1.786895147\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
good3 \( 1 + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.415 - 0.909i)T^{2} \)
31 \( 1 + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
41 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.142 - 0.989i)T^{2} \)
47 \( 1 + 0.830T + T^{2} \)
53 \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 + 0.989i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.959 - 0.281i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52118602894526312465367060057, −10.03422531446228157841042061379, −8.062542062027950652387126185099, −7.61138743437659244531148211816, −6.82550367212814907402543512137, −5.71918900608104115080657863044, −4.84065215965879831143058369525, −3.95151796193621052667428639418, −2.89329631811315640027675377386, −1.78704433341634774141698120543, 2.06946452549073772201552906675, 2.86604406608540997709264210580, 4.46141581877536034611072731035, 5.15489322225468777020878536551, 5.69866011077455003850553842204, 6.66773106891780830412046025579, 8.014464642242211627107600968312, 8.493203737147313875795545578468, 9.242724630853697542139753408197, 10.53718613730432873603078626101

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