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Properties

Label 2-1045-1045.398-c0-0-1
Degree $2$
Conductor $1045$
Sign $0.358 + 0.933i$
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  + (0.587 − 0.809i)4-s + (0.309 − 0.951i)5-s + (−0.0489 − 0.309i)7-s + (0.951 + 0.309i)9-s + (−0.587 − 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.809 + 1.58i)17-s + (−0.809 + 0.587i)19-s + (−0.587 − 0.809i)20-s + (0.642 + 0.642i)23-s + (−0.809 − 0.587i)25-s + (−0.278 − 0.142i)28-s + (−0.309 − 0.0489i)35-s + (0.809 − 0.587i)36-s + (0.642 − 0.642i)43-s − 44-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)4-s + (0.309 − 0.951i)5-s + (−0.0489 − 0.309i)7-s + (0.951 + 0.309i)9-s + (−0.587 − 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.809 + 1.58i)17-s + (−0.809 + 0.587i)19-s + (−0.587 − 0.809i)20-s + (0.642 + 0.642i)23-s + (−0.809 − 0.587i)25-s + (−0.278 − 0.142i)28-s + (−0.309 − 0.0489i)35-s + (0.809 − 0.587i)36-s + (0.642 − 0.642i)43-s − 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.219980594\)
\(L(\frac12)\) \(\approx\) \(1.219980594\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.587 + 0.809i)T^{2} \)
3 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \)
23 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
47 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.587 - 0.809i)T^{2} \)
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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

−10.15248602048790328984899510781, −9.185868601154882831758006481936, −8.345508273013926718676524038736, −7.47813377687806320466773640524, −6.39033716586705985720947513514, −5.76266142239538203902212655924, −4.81830302595688074196095682083, −3.88961080248027901762942480075, −2.20167155601630569231964382076, −1.26592378844397844517887311359, 2.21840498176827732095871074766, 2.75210331790125772987230562119, 4.02053273824161051808262000806, 4.99926436496561040808623809075, 6.43048229034877666049180617951, 7.01616405993595120342849693047, 7.43091491781221145469942124966, 8.641527678180693672532225525360, 9.519836070997721935022414381710, 10.33714128283168417128283868339

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