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Properties

Label 2-453-453.20-c0-0-0
Degree $2$
Conductor $453$
Sign $0.845 + 0.533i$
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  + (0.968 + 0.248i)3-s + (−0.809 − 0.587i)4-s + (0.348 − 1.82i)7-s + (0.876 + 0.481i)9-s + (−0.637 − 0.770i)12-s + (−1.62 + 0.645i)13-s + (0.309 + 0.951i)16-s + (0.688 + 0.500i)19-s + (0.791 − 1.68i)21-s + (0.535 + 0.844i)25-s + (0.728 + 0.684i)27-s + (−1.35 + 1.27i)28-s + (0.371 + 0.0469i)31-s + (−0.425 − 0.904i)36-s + (−1.80 + 0.713i)37-s + ⋯
L(s)  = 1  + (0.968 + 0.248i)3-s + (−0.809 − 0.587i)4-s + (0.348 − 1.82i)7-s + (0.876 + 0.481i)9-s + (−0.637 − 0.770i)12-s + (−1.62 + 0.645i)13-s + (0.309 + 0.951i)16-s + (0.688 + 0.500i)19-s + (0.791 − 1.68i)21-s + (0.535 + 0.844i)25-s + (0.728 + 0.684i)27-s + (−1.35 + 1.27i)28-s + (0.371 + 0.0469i)31-s + (−0.425 − 0.904i)36-s + (−1.80 + 0.713i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9856608580\)
\(L(\frac12)\) \(\approx\) \(0.9856608580\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.968 - 0.248i)T \)
151 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.535 - 0.844i)T^{2} \)
7 \( 1 + (-0.348 + 1.82i)T + (-0.929 - 0.368i)T^{2} \)
11 \( 1 + (-0.876 + 0.481i)T^{2} \)
13 \( 1 + (1.62 - 0.645i)T + (0.728 - 0.684i)T^{2} \)
17 \( 1 + (0.929 - 0.368i)T^{2} \)
19 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.992 + 0.125i)T^{2} \)
31 \( 1 + (-0.371 - 0.0469i)T + (0.968 + 0.248i)T^{2} \)
37 \( 1 + (1.80 - 0.713i)T + (0.728 - 0.684i)T^{2} \)
41 \( 1 + (0.425 + 0.904i)T^{2} \)
43 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
47 \( 1 + (0.425 + 0.904i)T^{2} \)
53 \( 1 + (0.187 + 0.982i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.598 + 0.153i)T + (0.876 - 0.481i)T^{2} \)
67 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
71 \( 1 + (0.929 + 0.368i)T^{2} \)
73 \( 1 + (-0.303 + 1.58i)T + (-0.929 - 0.368i)T^{2} \)
79 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
83 \( 1 + (0.637 + 0.770i)T^{2} \)
89 \( 1 + (0.637 + 0.770i)T^{2} \)
97 \( 1 + (-0.541 + 0.297i)T + (0.535 - 0.844i)T^{2} \)
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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

−10.82999977983433398537784773703, −10.07627399886954081158952246426, −9.636685688887504771113110556697, −8.582750332563762616595703711426, −7.54145471134431530847951965860, −6.95932935060834329385269967883, −5.05578315842562824004733741165, −4.43264558870637412839202845055, −3.43202387548862539154623412792, −1.54448663124299700064814488540, 2.37215847792831890311071634457, 3.11031893065669380077745523286, 4.66825033093199133663121293061, 5.45708671223543626538538002387, 7.06964267456176411432321712492, 7.993182616294991788771077174447, 8.685472078829080261567703427031, 9.305426866817933584078716102020, 10.10755130003660714696545471858, 11.83536465702861982416616191814

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