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Properties

Label 2-1045-1045.664-c0-0-4
Degree $2$
Conductor $1045$
Sign $-0.958 - 0.286i$
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  + (1.44 + 1.04i)2-s + (−0.610 + 1.87i)3-s + (0.672 + 2.06i)4-s + (0.809 − 0.587i)5-s + (−2.84 + 2.06i)6-s + (−0.647 + 1.99i)8-s + (−2.34 − 1.70i)9-s + 1.78·10-s + (0.587 − 0.809i)11-s − 4.29·12-s + (−0.734 − 0.533i)13-s + (0.610 + 1.87i)15-s + (−1.26 + 0.915i)16-s + (−1.59 − 4.91i)18-s + (−0.309 + 0.951i)19-s + (1.76 + 1.27i)20-s + ⋯
L(s)  = 1  + (1.44 + 1.04i)2-s + (−0.610 + 1.87i)3-s + (0.672 + 2.06i)4-s + (0.809 − 0.587i)5-s + (−2.84 + 2.06i)6-s + (−0.647 + 1.99i)8-s + (−2.34 − 1.70i)9-s + 1.78·10-s + (0.587 − 0.809i)11-s − 4.29·12-s + (−0.734 − 0.533i)13-s + (0.610 + 1.87i)15-s + (−1.26 + 0.915i)16-s + (−1.59 − 4.91i)18-s + (−0.309 + 0.951i)19-s + (1.76 + 1.27i)20-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.943410692\)
\(L(\frac12)\) \(\approx\) \(1.943410692\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - 0.312T + T^{2} \)
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 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.45277208926506780091999894913, −9.690759053367709369278055923190, −8.869823595817542619453245157099, −8.026426839010177935249416871962, −6.45136529589330533882193766690, −5.91756902027327805839341783655, −5.28766566851607684625130094348, −4.64346463384561226370745026063, −3.83916890617886800323516074630, −2.97744711335611325606335434421, 1.52602046161998704417753219774, 2.17888719356151531130453089708, 2.97784252467086174116369491528, 4.66117886265906752647743195255, 5.40329377929443662884401005786, 6.43699929303890304468733776581, 6.67397968100158297067532053336, 7.61122763799336124672405249612, 9.139145361290705017524811479129, 10.16893293215310327436163265126

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