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Properties

Label 2-1045-1045.949-c0-0-0
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: $\chi_{1045} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1496234450\)
\(L(\frac12)\) \(\approx\) \(0.1496234450\)
\(L(1)\) 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

−10.28503541272813345922590626271, −9.282886154444247034825662371706, −8.234055079200927962150816342179, −7.87372224776992685947850624642, −7.22958987224353223406111757089, −6.54233629253998208464116970160, −5.35214888966875717031487586100, −4.33395864518159594837999363340, −2.97536579047653453007591549812, −1.47472913302068063149451516290, 0.17732137181966874075500306757, 2.65286351502114337872849611534, 3.60254345950270701183754070118, 4.76538053045065270177955006570, 5.27707086646491639640642140796, 6.27380657632925068812619422060, 7.56967414010900847897286806100, 8.517006005291383123857067777824, 9.336563749736031395830184971188, 10.02671736232880949269033872661

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