Properties

Label 2-1045-1045.664-c0-0-8
Degree $2$
Conductor $1045$
Sign $-0.958 - 0.286i$
Analytic cond. $0.521522$
Root an. cond. $0.722165$
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  + (−1.44 − 1.04i)2-s + (0.610 − 1.87i)3-s + (0.672 + 2.06i)4-s + (0.809 − 0.587i)5-s + (−2.84 + 2.06i)6-s + (0.647 − 1.99i)8-s + (−2.34 − 1.70i)9-s − 1.78·10-s + (0.587 − 0.809i)11-s + 4.29·12-s + (0.734 + 0.533i)13-s + (−0.610 − 1.87i)15-s + (−1.26 + 0.915i)16-s + (1.59 + 4.91i)18-s + (−0.309 + 0.951i)19-s + (1.76 + 1.27i)20-s + ⋯
L(s)  = 1  + (−1.44 − 1.04i)2-s + (0.610 − 1.87i)3-s + (0.672 + 2.06i)4-s + (0.809 − 0.587i)5-s + (−2.84 + 2.06i)6-s + (0.647 − 1.99i)8-s + (−2.34 − 1.70i)9-s − 1.78·10-s + (0.587 − 0.809i)11-s + 4.29·12-s + (0.734 + 0.533i)13-s + (−0.610 − 1.87i)15-s + (−1.26 + 0.915i)16-s + (1.59 + 4.91i)18-s + (−0.309 + 0.951i)19-s + (1.76 + 1.27i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.958 - 0.286i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.958 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7066133479\)
\(L(\frac12)\) \(\approx\) \(0.7066133479\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.14 - 0.831i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + 0.312T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.269812594451553640083320164805, −8.872437230170308343625550313632, −8.294104485973668253351857951159, −7.57963737189701711271924647994, −6.50921323256115100953490343443, −5.92248301919790689938624180097, −3.65941188753594749780467080988, −2.62399262297717839404203554524, −1.66805489594990129007048250600, −1.08260855800417370324405127814, 2.12808408956816070720868881180, 3.40950060574572737892135438508, 4.71099667853622911171849857489, 5.60801870163851261407245051165, 6.42498541121277312613433111922, 7.44406017261633096681383263836, 8.477552448699167575265681841089, 9.005083366783088723760177156228, 9.646281750702577726369075837152, 10.14366572818759065613228850450

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