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Properties

Label 2-453-453.383-c0-0-0
Degree $2$
Conductor $453$
Sign $0.438 - 0.898i$
Analytic cond. $0.226076$
Root an. cond. $0.475474$
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  + (−0.992 + 0.125i)3-s + (0.309 + 0.951i)4-s + (0.238 − 0.288i)7-s + (0.968 − 0.248i)9-s + (−0.425 − 0.904i)12-s + (−0.362 + 1.90i)13-s + (−0.809 + 0.587i)16-s + (0.331 + 1.01i)19-s + (−0.200 + 0.316i)21-s + (0.876 − 0.481i)25-s + (−0.929 + 0.368i)27-s + (0.348 + 0.137i)28-s + (−0.0800 − 1.27i)31-s + (0.535 + 0.844i)36-s + (0.371 − 1.94i)37-s + ⋯
L(s)  = 1  + (−0.992 + 0.125i)3-s + (0.309 + 0.951i)4-s + (0.238 − 0.288i)7-s + (0.968 − 0.248i)9-s + (−0.425 − 0.904i)12-s + (−0.362 + 1.90i)13-s + (−0.809 + 0.587i)16-s + (0.331 + 1.01i)19-s + (−0.200 + 0.316i)21-s + (0.876 − 0.481i)25-s + (−0.929 + 0.368i)27-s + (0.348 + 0.137i)28-s + (−0.0800 − 1.27i)31-s + (0.535 + 0.844i)36-s + (0.371 − 1.94i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6910165253\)
\(L(\frac12)\) \(\approx\) \(0.6910165253\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.992 - 0.125i)T \)
151 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.876 + 0.481i)T^{2} \)
7 \( 1 + (-0.238 + 0.288i)T + (-0.187 - 0.982i)T^{2} \)
11 \( 1 + (-0.968 - 0.248i)T^{2} \)
13 \( 1 + (0.362 - 1.90i)T + (-0.929 - 0.368i)T^{2} \)
17 \( 1 + (0.187 - 0.982i)T^{2} \)
19 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.0627 - 0.998i)T^{2} \)
31 \( 1 + (0.0800 + 1.27i)T + (-0.992 + 0.125i)T^{2} \)
37 \( 1 + (-0.371 + 1.94i)T + (-0.929 - 0.368i)T^{2} \)
41 \( 1 + (-0.535 - 0.844i)T^{2} \)
43 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
47 \( 1 + (-0.535 - 0.844i)T^{2} \)
53 \( 1 + (0.637 + 0.770i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.60 - 0.202i)T + (0.968 + 0.248i)T^{2} \)
67 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
71 \( 1 + (0.187 + 0.982i)T^{2} \)
73 \( 1 + (0.393 - 0.476i)T + (-0.187 - 0.982i)T^{2} \)
79 \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \)
83 \( 1 + (0.425 + 0.904i)T^{2} \)
89 \( 1 + (0.425 + 0.904i)T^{2} \)
97 \( 1 + (1.56 + 0.402i)T + (0.876 + 0.481i)T^{2} \)
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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

−11.56507852696915415611613038352, −10.81941365199345370925794058483, −9.740851724569345144382437078614, −8.785300371761731428070837780196, −7.52177034217415947586606149256, −6.92939364803016289859999394465, −5.92748265513883054386390851136, −4.53945913140565451170679427922, −3.85549541024332553071687892107, −2.01181561364232635277688278676, 1.14210922338534384048802664827, 2.88342873760545828182090348197, 4.98508296101666480096266231959, 5.25618991093025610720020538789, 6.39620030501217223738285402257, 7.18984521734850350211079500297, 8.374030427096122304639869721020, 9.725691845042443668108292545326, 10.35943229060272807875596497400, 11.09166892539255595562885011300

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