Properties

Label 2-920-920.259-c0-0-1
Degree $2$
Conductor $920$
Sign $-0.117 + 0.993i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.118 + 0.822i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (−0.239 − 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (0.841 − 0.540i)18-s + (1.25 + 1.45i)19-s + (−0.959 − 0.281i)20-s − 1.30·22-s + (−0.959 − 0.281i)23-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.118 + 0.822i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (−0.239 − 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (0.841 − 0.540i)18-s + (1.25 + 1.45i)19-s + (−0.959 − 0.281i)20-s − 1.30·22-s + (−0.959 − 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291058075\)
\(L(\frac12)\) \(\approx\) \(1.291058075\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
11 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + 0.284T + T^{2} \)
53 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−10.20319185007286293180112835216, −9.549555797959036775065284840322, −8.472112882197336248335172536126, −7.87440758416569947212738162012, −6.03134693695241862177977687681, −5.59409455726080558168881956070, −4.90833150765758889822241213022, −3.46171556743462380009254772990, −2.55588980427917629181650139727, −1.32563768468995406613137229090, 1.99096557536213872472456854037, 3.48805675900816268410049136050, 4.47671088188292422564696184181, 5.21497456682079929288776634136, 6.58613009174313752327583325132, 6.98767708829989954818279294400, 7.46458899655945098744702046754, 8.945328428640206388884698155169, 9.752521860039209790105001849014, 10.04508802978105864823515751546

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