Properties

Label 2-1045-1045.284-c0-0-8
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  + (−0.0966 + 0.297i)2-s + (1.44 − 1.04i)3-s + (0.729 + 0.530i)4-s + (−0.309 − 0.951i)5-s + (0.172 + 0.530i)6-s + (−0.481 + 0.349i)8-s + (0.672 − 2.06i)9-s + 0.312·10-s + (−0.951 − 0.309i)11-s + 1.60·12-s + (−0.610 + 1.87i)13-s + (−1.44 − 1.04i)15-s + (0.221 + 0.680i)16-s + (0.550 + 0.400i)18-s + (0.809 − 0.587i)19-s + (0.278 − 0.857i)20-s + ⋯
L(s)  = 1  + (−0.0966 + 0.297i)2-s + (1.44 − 1.04i)3-s + (0.729 + 0.530i)4-s + (−0.309 − 0.951i)5-s + (0.172 + 0.530i)6-s + (−0.481 + 0.349i)8-s + (0.672 − 2.06i)9-s + 0.312·10-s + (−0.951 − 0.309i)11-s + 1.60·12-s + (−0.610 + 1.87i)13-s + (−1.44 − 1.04i)15-s + (0.221 + 0.680i)16-s + (0.550 + 0.400i)18-s + (0.809 − 0.587i)19-s + (0.278 − 0.857i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.601707506\)
\(L(\frac12)\) \(\approx\) \(1.601707506\)
\(L(1)\) 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 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + 0.907T + T^{2} \)
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 \( 1 + (-0.0966 + 0.297i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.480697769801395551483617134084, −8.984773478167463834883819787619, −8.286760070362749722851911535171, −7.49328675340264968820673108079, −7.18261516633785606741438118937, −6.11555261953621001051012350383, −4.69990473911418829559275618616, −3.51326488773457440377159413013, −2.53813625316108690803939248694, −1.66268835799979174627563984981, 2.20755853854052486369819765413, 3.00901781819505191858850658289, 3.43545127897419064151869777948, 4.91838525998672069329015057856, 5.77389864200690230592362813246, 7.27550969686240609691261226034, 7.70152215779545008142666836817, 8.537190168534896605772195296610, 9.818780935666514845394682619348, 10.20257237821876434742693938324

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