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Properties

Label 2-453-453.245-c0-0-0
Degree $2$
Conductor $453$
Sign $0.647 + 0.762i$
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  + (0.876 − 0.481i)3-s + (0.309 − 0.951i)4-s + (−1.35 + 0.536i)7-s + (0.535 − 0.844i)9-s + (−0.187 − 0.982i)12-s + (0.781 + 0.733i)13-s + (−0.809 − 0.587i)16-s + (−0.393 + 1.21i)19-s + (−0.929 + 1.12i)21-s + (−0.425 − 0.904i)25-s + (0.0627 − 0.998i)27-s + (0.0915 + 1.45i)28-s + (−1.80 + 0.462i)31-s + (−0.637 − 0.770i)36-s + (1.27 + 1.19i)37-s + ⋯
L(s)  = 1  + (0.876 − 0.481i)3-s + (0.309 − 0.951i)4-s + (−1.35 + 0.536i)7-s + (0.535 − 0.844i)9-s + (−0.187 − 0.982i)12-s + (0.781 + 0.733i)13-s + (−0.809 − 0.587i)16-s + (−0.393 + 1.21i)19-s + (−0.929 + 1.12i)21-s + (−0.425 − 0.904i)25-s + (0.0627 − 0.998i)27-s + (0.0915 + 1.45i)28-s + (−1.80 + 0.462i)31-s + (−0.637 − 0.770i)36-s + (1.27 + 1.19i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.647 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.065205334\)
\(L(\frac12)\) \(\approx\) \(1.065205334\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.876 + 0.481i)T \)
151 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.425 + 0.904i)T^{2} \)
7 \( 1 + (1.35 - 0.536i)T + (0.728 - 0.684i)T^{2} \)
11 \( 1 + (-0.535 - 0.844i)T^{2} \)
13 \( 1 + (-0.781 - 0.733i)T + (0.0627 + 0.998i)T^{2} \)
17 \( 1 + (-0.728 - 0.684i)T^{2} \)
19 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.968 + 0.248i)T^{2} \)
31 \( 1 + (1.80 - 0.462i)T + (0.876 - 0.481i)T^{2} \)
37 \( 1 + (-1.27 - 1.19i)T + (0.0627 + 0.998i)T^{2} \)
41 \( 1 + (0.637 + 0.770i)T^{2} \)
43 \( 1 + (-1.84 - 0.730i)T + (0.728 + 0.684i)T^{2} \)
47 \( 1 + (0.637 + 0.770i)T^{2} \)
53 \( 1 + (0.929 + 0.368i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.41 + 0.779i)T + (0.535 + 0.844i)T^{2} \)
67 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
71 \( 1 + (-0.728 + 0.684i)T^{2} \)
73 \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \)
79 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
83 \( 1 + (0.187 + 0.982i)T^{2} \)
89 \( 1 + (0.187 + 0.982i)T^{2} \)
97 \( 1 + (0.866 + 1.36i)T + (-0.425 + 0.904i)T^{2} \)
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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

−11.11570467341152462757050372380, −10.01125731415705635672337496072, −9.421313423743860224773692095612, −8.653041543680796385481717896319, −7.42294604381942374905902090037, −6.29651965843252409148304439315, −6.01993701382360234518488416471, −4.14211814936660247118845241270, −2.92887208714141529370633402191, −1.70480447967165128942087568446, 2.54685136610905343252326267959, 3.48439810287221572545688217463, 4.14520725011323709515402418463, 5.87130669174632693943243415897, 7.15464260749697017894812199418, 7.66302107219943780087975742385, 8.932864282457912319281626108918, 9.344606381989913862285302880975, 10.59178923617461593722267311168, 11.16075814220615742094847888418

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