Properties

Label 2-1045-209.208-c1-0-7
Degree $2$
Conductor $1045$
Sign $0.740 - 0.672i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.512·2-s − 3.19i·3-s − 1.73·4-s − 5-s − 1.64i·6-s + 1.77i·7-s − 1.91·8-s − 7.23·9-s − 0.512·10-s + (−0.881 + 3.19i)11-s + 5.55i·12-s + 2.41·13-s + 0.909i·14-s + 3.19i·15-s + 2.49·16-s − 1.90i·17-s + ⋯
L(s)  = 1  + 0.362·2-s − 1.84i·3-s − 0.868·4-s − 0.447·5-s − 0.669i·6-s + 0.669i·7-s − 0.677·8-s − 2.41·9-s − 0.162·10-s + (−0.265 + 0.964i)11-s + 1.60i·12-s + 0.669·13-s + 0.242i·14-s + 0.825i·15-s + 0.622·16-s − 0.462i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.740 - 0.672i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.740 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6365569633\)
\(L(\frac12)\) \(\approx\) \(0.6365569633\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + (0.881 - 3.19i)T \)
19 \( 1 + (-3.68 - 2.33i)T \)
good2 \( 1 - 0.512T + 2T^{2} \)
3 \( 1 + 3.19iT - 3T^{2} \)
7 \( 1 - 1.77iT - 7T^{2} \)
13 \( 1 - 2.41T + 13T^{2} \)
17 \( 1 + 1.90iT - 17T^{2} \)
23 \( 1 + 3.67T + 23T^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 + 0.955iT - 31T^{2} \)
37 \( 1 - 6.90iT - 37T^{2} \)
41 \( 1 - 0.482T + 41T^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 - 0.519T + 47T^{2} \)
53 \( 1 - 2.92iT - 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + 7.83iT - 61T^{2} \)
67 \( 1 - 3.84iT - 67T^{2} \)
71 \( 1 - 0.976iT - 71T^{2} \)
73 \( 1 - 0.0771iT - 73T^{2} \)
79 \( 1 + 7.50T + 79T^{2} \)
83 \( 1 - 2.81iT - 83T^{2} \)
89 \( 1 - 15.4iT - 89T^{2} \)
97 \( 1 - 6.02iT - 97T^{2} \)
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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

−9.771162472968351537926891504575, −9.013864517825326594874633530840, −8.012988782119217314941348423605, −7.73947003978978580138432220011, −6.61104208826699122779090557275, −5.81619761219185476012838203827, −5.02838260423712016433542107150, −3.65677923059828225333086061125, −2.53910746759556114445787090979, −1.29556183000666905987768579678, 0.28489012145156323822903982313, 3.14566288217542626285494916283, 3.85293555615216729356235422503, 4.24617293483051347976437838068, 5.43075143650850500668739440409, 5.80185463090565053270647310654, 7.51027833374781958132120901781, 8.632596219884299744357253357437, 8.933486251620699879571419602775, 9.892615833089893706751486759936

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