Properties

Label 2-453-453.29-c0-0-0
Degree $2$
Conductor $453$
Sign $0.311 - 0.950i$
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  + (−0.187 + 0.982i)3-s + (0.309 + 0.951i)4-s + (1.69 − 0.435i)7-s + (−0.929 − 0.368i)9-s + (−0.992 + 0.125i)12-s + (−1.62 − 0.895i)13-s + (−0.809 + 0.587i)16-s + (0.0388 + 0.119i)19-s + (0.110 + 1.74i)21-s + (0.728 + 0.684i)25-s + (0.535 − 0.844i)27-s + (0.939 + 1.47i)28-s + (−1.23 − 1.49i)31-s + (0.0627 − 0.998i)36-s + (−0.328 − 0.180i)37-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)3-s + (0.309 + 0.951i)4-s + (1.69 − 0.435i)7-s + (−0.929 − 0.368i)9-s + (−0.992 + 0.125i)12-s + (−1.62 − 0.895i)13-s + (−0.809 + 0.587i)16-s + (0.0388 + 0.119i)19-s + (0.110 + 1.74i)21-s + (0.728 + 0.684i)25-s + (0.535 − 0.844i)27-s + (0.939 + 1.47i)28-s + (−1.23 − 1.49i)31-s + (0.0627 − 0.998i)36-s + (−0.328 − 0.180i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9194745362\)
\(L(\frac12)\) \(\approx\) \(0.9194745362\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.187 - 0.982i)T \)
151 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.728 - 0.684i)T^{2} \)
7 \( 1 + (-1.69 + 0.435i)T + (0.876 - 0.481i)T^{2} \)
11 \( 1 + (0.929 - 0.368i)T^{2} \)
13 \( 1 + (1.62 + 0.895i)T + (0.535 + 0.844i)T^{2} \)
17 \( 1 + (-0.876 - 0.481i)T^{2} \)
19 \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.637 + 0.770i)T^{2} \)
31 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
37 \( 1 + (0.328 + 0.180i)T + (0.535 + 0.844i)T^{2} \)
41 \( 1 + (-0.0627 + 0.998i)T^{2} \)
43 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
47 \( 1 + (-0.0627 + 0.998i)T^{2} \)
53 \( 1 + (-0.968 - 0.248i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.303 - 1.58i)T + (-0.929 + 0.368i)T^{2} \)
67 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
71 \( 1 + (-0.876 + 0.481i)T^{2} \)
73 \( 1 + (-0.598 + 0.153i)T + (0.876 - 0.481i)T^{2} \)
79 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
83 \( 1 + (0.992 - 0.125i)T^{2} \)
89 \( 1 + (0.992 - 0.125i)T^{2} \)
97 \( 1 + (-1.50 + 0.595i)T + (0.728 - 0.684i)T^{2} \)
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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

−11.35554635179758645903182897446, −10.77274488896630079813755065424, −9.782688620188235773230093608296, −8.668926262386278208592422676884, −7.85160877369549239595562156120, −7.18551477364581972225505368721, −5.43966536860211811159081041233, −4.72884897902447948525066446093, −3.71597531136328624709975116924, −2.38207654027055214659021692046, 1.59654921021572787441263118420, 2.38464112699019949780195248097, 4.89816415683581011871239378575, 5.23418503103595976536744316554, 6.60917444140040560921785982401, 7.29216041204330865167419584596, 8.310760568806795300790670984177, 9.226469027443271036353700275322, 10.46142256910348050449046310618, 11.29142396929123064323416227233

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