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Properties

Label 2-469-469.370-c0-0-0
Degree $2$
Conductor $469$
Sign $0.755 + 0.655i$
Analytic cond. $0.234061$
Root an. cond. $0.483799$
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  + (0.195 − 0.807i)2-s + (0.275 + 0.141i)4-s + (0.723 + 0.690i)7-s + (0.712 − 0.822i)8-s + (−0.959 + 0.281i)9-s + (−0.469 + 0.0448i)11-s + (0.698 − 0.449i)14-s + (−0.344 − 0.484i)16-s + (0.0395 + 0.829i)18-s + (−0.0557 + 0.387i)22-s + (−0.607 − 0.243i)23-s + (−0.654 − 0.755i)25-s + (0.101 + 0.292i)28-s + (−0.723 − 1.25i)29-s + (0.551 − 0.220i)32-s + ⋯
L(s)  = 1  + (0.195 − 0.807i)2-s + (0.275 + 0.141i)4-s + (0.723 + 0.690i)7-s + (0.712 − 0.822i)8-s + (−0.959 + 0.281i)9-s + (−0.469 + 0.0448i)11-s + (0.698 − 0.449i)14-s + (−0.344 − 0.484i)16-s + (0.0395 + 0.829i)18-s + (−0.0557 + 0.387i)22-s + (−0.607 − 0.243i)23-s + (−0.654 − 0.755i)25-s + (0.101 + 0.292i)28-s + (−0.723 − 1.25i)29-s + (0.551 − 0.220i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(0.234061\)
Root analytic conductor: \(0.483799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 469,\ (\ :0),\ 0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.067038420\)
\(L(\frac12)\) \(\approx\) \(1.067038420\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
good2 \( 1 + (-0.195 + 0.807i)T + (-0.888 - 0.458i)T^{2} \)
3 \( 1 + (0.959 - 0.281i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
11 \( 1 + (0.469 - 0.0448i)T + (0.981 - 0.189i)T^{2} \)
13 \( 1 + (-0.928 + 0.371i)T^{2} \)
17 \( 1 + (-0.580 + 0.814i)T^{2} \)
19 \( 1 + (-0.0475 + 0.998i)T^{2} \)
23 \( 1 + (0.607 + 0.243i)T + (0.723 + 0.690i)T^{2} \)
29 \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.928 - 0.371i)T^{2} \)
37 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.995 + 0.0950i)T^{2} \)
43 \( 1 + (-0.0800 - 0.0514i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.235 + 0.971i)T^{2} \)
53 \( 1 + (1.32 - 0.849i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.981 - 0.189i)T^{2} \)
71 \( 1 + (-1.76 - 0.912i)T + (0.580 + 0.814i)T^{2} \)
73 \( 1 + (-0.981 - 0.189i)T^{2} \)
79 \( 1 + (-0.0930 + 0.268i)T + (-0.786 - 0.618i)T^{2} \)
83 \( 1 + (0.327 - 0.945i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
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

−11.33625883601456186547812244608, −10.51123602717438167163084840950, −9.557806925393431309618562746711, −8.295806869470953299002924715930, −7.79177551063910348194586103685, −6.36484926599933308818632403937, −5.36792619700455576663944737922, −4.19714370065163223901255961279, −2.83162428020798949859095837238, −2.00959806934931622663597479177, 1.93060646115033413893794571215, 3.57207239182446973499791038418, 5.02308561766494075526154224309, 5.67484844449312626197197593720, 6.76644583681656313923597228787, 7.66025382689750038748192874875, 8.290167586209870143724611614999, 9.470044871687967471096783123401, 10.78225871149481631181486792683, 11.10379014341459761548955194094

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