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Properties

Label 2-1045-1045.664-c0-0-9
Degree $2$
Conductor $1045$
Sign $-0.286 + 0.958i$
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  + (0.5 + 0.363i)2-s + (0.5 − 1.53i)3-s + (−0.190 − 0.587i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−1.30 − 0.951i)9-s − 0.618·10-s + (−0.809 − 0.587i)11-s − 12-s + (0.5 + 0.363i)13-s + (0.5 + 1.53i)15-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.5 + 0.363i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.363i)2-s + (0.5 − 1.53i)3-s + (−0.190 − 0.587i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−1.30 − 0.951i)9-s − 0.618·10-s + (−0.809 − 0.587i)11-s − 12-s + (0.5 + 0.363i)13-s + (0.5 + 1.53i)15-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.5 + 0.363i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.286 + 0.958i$
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: \((2,\ 1045,\ (\ :0),\ -0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198571997\)
\(L(\frac12)\) \(\approx\) \(1.198571997\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - 1.61T + T^{2} \)
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 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.847301922104327091171063081802, −8.686776005902690497707155373100, −8.044013992557834281487262072729, −7.17956378520416800575798428200, −6.65195601822996493768912045540, −5.87189191028273826929244737667, −4.71536712394089586724552567149, −3.46548776312403070719615115986, −2.47685343319124064022258699028, −0.950504374774299536713271929731, 2.46250025927112361202436008661, 3.66140989280764588052589120736, 3.90919550956115731307420495052, 4.93453785578919682234408112357, 5.45917042125901791129522921265, 7.37971833560627100841222513019, 8.180380394816773625685766847038, 8.646014186531621616246005611078, 9.584684134114830841235381236687, 10.37096003960788698513573144638

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