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Properties

Label 2-1045-209.208-c1-0-0
Degree $2$
Conductor $1045$
Sign $-0.801 + 0.597i$
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  − 1.55·2-s + 2.37i·3-s + 0.426·4-s − 5-s − 3.69i·6-s − 4.79i·7-s + 2.45·8-s − 2.63·9-s + 1.55·10-s + (0.945 + 3.17i)11-s + 1.01i·12-s + 2.33·13-s + 7.46i·14-s − 2.37i·15-s − 4.67·16-s + 3.56i·17-s + ⋯
L(s)  = 1  − 1.10·2-s + 1.37i·3-s + 0.213·4-s − 0.447·5-s − 1.51i·6-s − 1.81i·7-s + 0.866·8-s − 0.879·9-s + 0.492·10-s + (0.285 + 0.958i)11-s + 0.292i·12-s + 0.646·13-s + 1.99i·14-s − 0.613i·15-s − 1.16·16-s + 0.864i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.801 + 0.597i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.801 + 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07161219072\)
\(L(\frac12)\) \(\approx\) \(0.07161219072\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.945 - 3.17i)T \)
19 \( 1 + (1.49 + 4.09i)T \)
good2 \( 1 + 1.55T + 2T^{2} \)
3 \( 1 - 2.37iT - 3T^{2} \)
7 \( 1 + 4.79iT - 7T^{2} \)
13 \( 1 - 2.33T + 13T^{2} \)
17 \( 1 - 3.56iT - 17T^{2} \)
23 \( 1 + 7.00T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 0.774iT - 31T^{2} \)
37 \( 1 - 4.42iT - 37T^{2} \)
41 \( 1 - 5.97T + 41T^{2} \)
43 \( 1 - 5.30iT - 43T^{2} \)
47 \( 1 + 5.52T + 47T^{2} \)
53 \( 1 - 7.05iT - 53T^{2} \)
59 \( 1 + 4.96iT - 59T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 + 8.09iT - 71T^{2} \)
73 \( 1 + 16.6iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 3.17iT - 83T^{2} \)
89 \( 1 - 6.42iT - 89T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−10.19975767909413365645332060428, −9.723617429751218213039236181883, −9.029218838390184815369679949646, −7.974074856546162271288210260805, −7.46210264182635917700745927287, −6.43926112812478432993644953237, −4.79751885865937744934772794925, −4.15345049165499641127894999136, −3.72858780004764927978873565590, −1.54617134501475357048366614070, 0.04986986170565458721176603906, 1.52067950592806942580343910773, 2.40809180868564306033934449246, 3.87439328240775016021211193698, 5.57702499243898727844460107251, 6.08560983941863607392073722737, 7.21661063941604380436423320279, 7.971762873533399601479355597542, 8.557040269109291057337362186223, 9.048263752235403182699767400010

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