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Properties

Label 2-453-453.242-c0-0-0
Degree $2$
Conductor $453$
Sign $0.113 - 0.993i$
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  + (0.535 + 0.844i)3-s + (−0.809 + 0.587i)4-s + (0.0915 + 0.0859i)7-s + (−0.425 + 0.904i)9-s + (−0.929 − 0.368i)12-s + (−0.0534 + 0.849i)13-s + (0.309 − 0.951i)16-s + (0.303 − 0.220i)19-s + (−0.0235 + 0.123i)21-s + (−0.637 − 0.770i)25-s + (−0.992 + 0.125i)27-s + (−0.124 − 0.0157i)28-s + (1.27 + 0.702i)31-s + (−0.187 − 0.982i)36-s + (0.0672 − 1.06i)37-s + ⋯
L(s)  = 1  + (0.535 + 0.844i)3-s + (−0.809 + 0.587i)4-s + (0.0915 + 0.0859i)7-s + (−0.425 + 0.904i)9-s + (−0.929 − 0.368i)12-s + (−0.0534 + 0.849i)13-s + (0.309 − 0.951i)16-s + (0.303 − 0.220i)19-s + (−0.0235 + 0.123i)21-s + (−0.637 − 0.770i)25-s + (−0.992 + 0.125i)27-s + (−0.124 − 0.0157i)28-s + (1.27 + 0.702i)31-s + (−0.187 − 0.982i)36-s + (0.0672 − 1.06i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.113 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8420554170\)
\(L(\frac12)\) \(\approx\) \(0.8420554170\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.535 - 0.844i)T \)
151 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.637 + 0.770i)T^{2} \)
7 \( 1 + (-0.0915 - 0.0859i)T + (0.0627 + 0.998i)T^{2} \)
11 \( 1 + (0.425 + 0.904i)T^{2} \)
13 \( 1 + (0.0534 - 0.849i)T + (-0.992 - 0.125i)T^{2} \)
17 \( 1 + (-0.0627 + 0.998i)T^{2} \)
19 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (-1.27 - 0.702i)T + (0.535 + 0.844i)T^{2} \)
37 \( 1 + (-0.0672 + 1.06i)T + (-0.992 - 0.125i)T^{2} \)
41 \( 1 + (0.187 + 0.982i)T^{2} \)
43 \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \)
47 \( 1 + (0.187 + 0.982i)T^{2} \)
53 \( 1 + (-0.728 + 0.684i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.331 + 0.521i)T + (-0.425 - 0.904i)T^{2} \)
67 \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \)
71 \( 1 + (-0.0627 - 0.998i)T^{2} \)
73 \( 1 + (1.17 + 1.10i)T + (0.0627 + 0.998i)T^{2} \)
79 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
83 \( 1 + (0.929 + 0.368i)T^{2} \)
89 \( 1 + (0.929 + 0.368i)T^{2} \)
97 \( 1 + (0.263 + 0.559i)T + (-0.637 + 0.770i)T^{2} \)
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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

−11.52963218193879693836558782354, −10.38691806939797260183395510486, −9.583961314939046184543017664226, −8.842607475687940953656051159007, −8.176754442321026604683345222558, −7.11973555720351329109714504214, −5.55575041293592220839169498797, −4.51246706364750270259636802725, −3.80834549875051587850015407758, −2.52841782061051431094780774400, 1.23788316917935426543115565974, 2.89989666068012529645773382489, 4.22562598066447790123546923183, 5.52409101550512827188679220265, 6.35046650653830496776320952394, 7.66612646960300760441419128496, 8.260676154111562820797025020683, 9.315723514544758138913952493947, 9.962106336475683494128054212103, 11.09830784158864981276522641128

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