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Properties

Label 2-1045-1045.607-c0-0-0
Degree $2$
Conductor $1045$
Sign $-0.995 - 0.0965i$
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 + (−1.95 + 0.309i)7-s + (−0.951 − 0.309i)9-s + (0.587 + 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.809 − 0.412i)17-s + (−0.809 + 0.587i)19-s + (0.587 + 0.809i)20-s + (−1.26 + 1.26i)23-s + (−0.809 − 0.587i)25-s + (0.896 − 1.76i)28-s + (−0.309 + 1.95i)35-s + (0.809 − 0.587i)36-s + (−1.26 − 1.26i)43-s − 44-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)4-s + (0.309 − 0.951i)5-s + (−1.95 + 0.309i)7-s + (−0.951 − 0.309i)9-s + (0.587 + 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.809 − 0.412i)17-s + (−0.809 + 0.587i)19-s + (0.587 + 0.809i)20-s + (−1.26 + 1.26i)23-s + (−0.809 − 0.587i)25-s + (0.896 − 1.76i)28-s + (−0.309 + 1.95i)35-s + (0.809 − 0.587i)36-s + (−1.26 − 1.26i)43-s − 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.995 - 0.0965i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.995 - 0.0965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09734514770\)
\(L(\frac12)\) \(\approx\) \(0.09734514770\)
\(L(1)\) 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 \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \)
23 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
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 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
47 \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (-0.221 - 1.39i)T + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
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

−10.09048074741266715868686105118, −9.491663732856344404405986576751, −8.992427582106624942392507835195, −8.332514808791495028267853456957, −7.12599713744283569406807591146, −6.24235317700236718532918357628, −5.46563069500010072617840018814, −4.18044667961218228684078735921, −3.52426926814187165538719840398, −2.31498609638475493110199832038, 0.083150517460411159550230369345, 2.36589506210677997749271771848, 3.34744496540404984192603111420, 4.30086756444118193526864531424, 5.81783977392462447073778840146, 6.33595366972273724228636543459, 6.69871727372951708731144185670, 8.295694573972434712433871733731, 9.084033886787562790454224073755, 9.732203308203873771180681326469

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