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Properties

Label 2-1045-1045.284-c0-0-2
Degree $2$
Conductor $1045$
Sign $-0.835 - 0.550i$
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  + (−0.610 + 1.87i)2-s + (−0.734 + 0.533i)3-s + (−2.34 − 1.70i)4-s + (−0.309 − 0.951i)5-s + (−0.554 − 1.70i)6-s + (3.03 − 2.20i)8-s + (−0.0542 + 0.166i)9-s + 1.97·10-s + (0.951 + 0.309i)11-s + 2.63·12-s + (−0.0966 + 0.297i)13-s + (0.734 + 0.533i)15-s + (1.39 + 4.29i)16-s + (−0.280 − 0.203i)18-s + (0.809 − 0.587i)19-s + (−0.896 + 2.76i)20-s + ⋯
L(s)  = 1  + (−0.610 + 1.87i)2-s + (−0.734 + 0.533i)3-s + (−2.34 − 1.70i)4-s + (−0.309 − 0.951i)5-s + (−0.554 − 1.70i)6-s + (3.03 − 2.20i)8-s + (−0.0542 + 0.166i)9-s + 1.97·10-s + (0.951 + 0.309i)11-s + 2.63·12-s + (−0.0966 + 0.297i)13-s + (0.734 + 0.533i)15-s + (1.39 + 4.29i)16-s + (−0.280 − 0.203i)18-s + (0.809 − 0.587i)19-s + (−0.896 + 2.76i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.835 - 0.550i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.835 - 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4765240479\)
\(L(\frac12)\) \(\approx\) \(0.4765240479\)
\(L(1)\) 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 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.78T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.931464863726757399983397876865, −9.437323606132599110434518960239, −8.730839837137981382390622567656, −7.902595586382944945738395359457, −7.15219630171757528686569131711, −6.18403987238733165529005526817, −5.48353584135087530690348775721, −4.65577273921059176165901475620, −4.20489879249271307184714112514, −1.09518782375882725800935842988, 0.812487694459108776906475400755, 2.12913614053807361852304908694, 3.35542498277372434127428858680, 3.88015444342422458962943319614, 5.30300273445489790697803331044, 6.50348631776941310630114689127, 7.51836731098317609863368447811, 8.298820272282845646531413506940, 9.385877815438238520201336806703, 9.899457829169754577956730997129

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