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Properties

Label 2-453-453.386-c0-0-0
Degree $2$
Conductor $453$
Sign $0.324 + 0.945i$
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  + (0.0627 − 0.998i)3-s + (−0.809 + 0.587i)4-s + (0.542 − 1.15i)7-s + (−0.992 − 0.125i)9-s + (0.535 + 0.844i)12-s + (1.26 − 1.52i)13-s + (0.309 − 0.951i)16-s + (−1.41 + 1.03i)19-s + (−1.11 − 0.614i)21-s + (0.968 + 0.248i)25-s + (−0.187 + 0.982i)27-s + (0.238 + 1.25i)28-s + (−0.620 + 0.582i)31-s + (0.876 − 0.481i)36-s + (−0.0800 + 0.0967i)37-s + ⋯
L(s)  = 1  + (0.0627 − 0.998i)3-s + (−0.809 + 0.587i)4-s + (0.542 − 1.15i)7-s + (−0.992 − 0.125i)9-s + (0.535 + 0.844i)12-s + (1.26 − 1.52i)13-s + (0.309 − 0.951i)16-s + (−1.41 + 1.03i)19-s + (−1.11 − 0.614i)21-s + (0.968 + 0.248i)25-s + (−0.187 + 0.982i)27-s + (0.238 + 1.25i)28-s + (−0.620 + 0.582i)31-s + (0.876 − 0.481i)36-s + (−0.0800 + 0.0967i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.324 + 0.945i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.324 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7597945607\)
\(L(\frac12)\) \(\approx\) \(0.7597945607\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0627 + 0.998i)T \)
151 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.968 - 0.248i)T^{2} \)
7 \( 1 + (-0.542 + 1.15i)T + (-0.637 - 0.770i)T^{2} \)
11 \( 1 + (0.992 - 0.125i)T^{2} \)
13 \( 1 + (-1.26 + 1.52i)T + (-0.187 - 0.982i)T^{2} \)
17 \( 1 + (0.637 - 0.770i)T^{2} \)
19 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.728 + 0.684i)T^{2} \)
31 \( 1 + (0.620 - 0.582i)T + (0.0627 - 0.998i)T^{2} \)
37 \( 1 + (0.0800 - 0.0967i)T + (-0.187 - 0.982i)T^{2} \)
41 \( 1 + (-0.876 + 0.481i)T^{2} \)
43 \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \)
47 \( 1 + (-0.876 + 0.481i)T^{2} \)
53 \( 1 + (0.425 + 0.904i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.0388 - 0.616i)T + (-0.992 + 0.125i)T^{2} \)
67 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
71 \( 1 + (0.637 + 0.770i)T^{2} \)
73 \( 1 + (-0.688 + 1.46i)T + (-0.637 - 0.770i)T^{2} \)
79 \( 1 + (-1.41 - 1.32i)T + (0.0627 + 0.998i)T^{2} \)
83 \( 1 + (-0.535 - 0.844i)T^{2} \)
89 \( 1 + (-0.535 - 0.844i)T^{2} \)
97 \( 1 + (0.613 - 0.0774i)T + (0.968 - 0.248i)T^{2} \)
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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

−10.98488901427627533348416026890, −10.50387106711602564032092193376, −9.000756087649356470138754973637, −8.105441474293616003166933149978, −7.80390057548606920886899235877, −6.59146784650960442785441472336, −5.46942738433752531334289739078, −4.14991760071383493053720319925, −3.14802822794219909218655949187, −1.16839593808152149693475151303, 2.16382140834274056598001907531, 3.92481558986725385537645444259, 4.69078239999992012791800891520, 5.62154911071145698600806117232, 6.51886619616511321630235835508, 8.505544483974954654136716830159, 8.843504407510025741865576649670, 9.417057270807076572975044487619, 10.73223432581872056818456458584, 11.12903873041659383478092371651

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