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Properties

Label 2-453-453.8-c0-0-0
Degree $2$
Conductor $453$
Sign $0.667 - 0.744i$
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  + (0.309 + 0.951i)3-s + 4-s + (−0.5 + 0.363i)7-s + (−0.809 + 0.587i)9-s + (0.309 + 0.951i)12-s + (−0.5 − 1.53i)13-s + 16-s − 1.61·19-s + (−0.5 − 0.363i)21-s + (0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.5 + 0.363i)28-s + (1.30 + 0.951i)31-s + (−0.809 + 0.587i)36-s + (0.190 + 0.587i)37-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + 4-s + (−0.5 + 0.363i)7-s + (−0.809 + 0.587i)9-s + (0.309 + 0.951i)12-s + (−0.5 − 1.53i)13-s + 16-s − 1.61·19-s + (−0.5 − 0.363i)21-s + (0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.5 + 0.363i)28-s + (1.30 + 0.951i)31-s + (−0.809 + 0.587i)36-s + (0.190 + 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046049634\)
\(L(\frac12)\) \(\approx\) \(1.046049634\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
151 \( 1 - T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
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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

−11.20515740257932664144700684006, −10.33292644895910635971380136339, −10.01226496869737960387614809059, −8.607012187352672559033252837455, −8.001610134448719383525699738573, −6.64986173560466505846186916034, −5.79901122405854388037817760458, −4.66652325175467379957800559067, −3.22614298414060537489477774952, −2.48615228190880590178462632677, 1.77080127303932958247840334988, 2.78110996084825282094531364197, 4.18206251790532487468604274879, 5.95549345536806626434465202062, 6.72873456588323820224302312627, 7.23832866953520692671593241660, 8.319507250542547999419866722245, 9.309260162767801236066341441552, 10.38115856590173117708828295792, 11.43129432405902341527264226527

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