Properties

Label 2-45-1.1-c3-0-1
Degree $2$
Conductor $45$
Sign $-1$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 17·4-s + 5·5-s − 30·7-s − 45·8-s − 25·10-s − 50·11-s − 20·13-s + 150·14-s + 89·16-s + 10·17-s − 44·19-s + 85·20-s + 250·22-s − 120·23-s + 25·25-s + 100·26-s − 510·28-s + 50·29-s + 108·31-s − 85·32-s − 50·34-s − 150·35-s − 40·37-s + 220·38-s − 225·40-s − 400·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s + 0.447·5-s − 1.61·7-s − 1.98·8-s − 0.790·10-s − 1.37·11-s − 0.426·13-s + 2.86·14-s + 1.39·16-s + 0.142·17-s − 0.531·19-s + 0.950·20-s + 2.42·22-s − 1.08·23-s + 1/5·25-s + 0.754·26-s − 3.44·28-s + 0.320·29-s + 0.625·31-s − 0.469·32-s − 0.252·34-s − 0.724·35-s − 0.177·37-s + 0.939·38-s − 0.889·40-s − 1.52·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{45} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 + 30 T + p^{3} T^{2} \)
11 \( 1 + 50 T + p^{3} T^{2} \)
13 \( 1 + 20 T + p^{3} T^{2} \)
17 \( 1 - 10 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 - 50 T + p^{3} T^{2} \)
31 \( 1 - 108 T + p^{3} T^{2} \)
37 \( 1 + 40 T + p^{3} T^{2} \)
41 \( 1 + 400 T + p^{3} T^{2} \)
43 \( 1 - 280 T + p^{3} T^{2} \)
47 \( 1 - 280 T + p^{3} T^{2} \)
53 \( 1 - 610 T + p^{3} T^{2} \)
59 \( 1 + 50 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 + 180 T + p^{3} T^{2} \)
71 \( 1 + 700 T + p^{3} T^{2} \)
73 \( 1 + 410 T + p^{3} T^{2} \)
79 \( 1 + 516 T + p^{3} T^{2} \)
83 \( 1 + 660 T + p^{3} T^{2} \)
89 \( 1 - 1500 T + p^{3} T^{2} \)
97 \( 1 + 1630 T + p^{3} T^{2} \)
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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

−15.45497212253526830584801893620, −13.43476142079945067747989618318, −12.19526613801690446144734492354, −10.38829604137491425371710973860, −9.987865419041304881742737081449, −8.742408380948106847041709722572, −7.37261708537653106302968142221, −6.10308818020358800413604681838, −2.60643207316796608487175214091, 0, 2.60643207316796608487175214091, 6.10308818020358800413604681838, 7.37261708537653106302968142221, 8.742408380948106847041709722572, 9.987865419041304881742737081449, 10.38829604137491425371710973860, 12.19526613801690446144734492354, 13.43476142079945067747989618318, 15.45497212253526830584801893620

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