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Properties

Label 2-1045-5.4-c1-0-46
Degree $2$
Conductor $1045$
Sign $0.987 + 0.154i$
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  + 0.379i·2-s − 2.60i·3-s + 1.85·4-s + (2.20 + 0.346i)5-s + 0.989·6-s + 4.15i·7-s + 1.46i·8-s − 3.79·9-s + (−0.131 + 0.837i)10-s + 11-s − 4.84i·12-s − 0.808i·13-s − 1.57·14-s + (0.903 − 5.76i)15-s + 3.15·16-s + 5.26i·17-s + ⋯
L(s)  = 1  + 0.268i·2-s − 1.50i·3-s + 0.928·4-s + (0.987 + 0.154i)5-s + 0.403·6-s + 1.57i·7-s + 0.517i·8-s − 1.26·9-s + (−0.0415 + 0.264i)10-s + 0.301·11-s − 1.39i·12-s − 0.224i·13-s − 0.421·14-s + (0.233 − 1.48i)15-s + 0.789·16-s + 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.987 + 0.154i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.441212603\)
\(L(\frac12)\) \(\approx\) \(2.441212603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.20 - 0.346i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 0.379iT - 2T^{2} \)
3 \( 1 + 2.60iT - 3T^{2} \)
7 \( 1 - 4.15iT - 7T^{2} \)
13 \( 1 + 0.808iT - 13T^{2} \)
17 \( 1 - 5.26iT - 17T^{2} \)
23 \( 1 + 2.49iT - 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + 2.11iT - 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 1.42iT - 43T^{2} \)
47 \( 1 - 4.28iT - 47T^{2} \)
53 \( 1 + 7.81iT - 53T^{2} \)
59 \( 1 - 9.86T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 + 7.58iT - 67T^{2} \)
71 \( 1 + 8.49T + 71T^{2} \)
73 \( 1 + 6.86iT - 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 - 1.09T + 89T^{2} \)
97 \( 1 - 9.76iT - 97T^{2} \)
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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

−9.921781825017512613501642219876, −8.623168103965297104740079249032, −8.325956749739494064672343282608, −7.17011123755745178018138929099, −6.28684430027120535807333665841, −6.15662533612772250798768294728, −5.22542011127888444442587105116, −3.02381376896340822372059131371, −2.16356385493529865809695019826, −1.60972102742721721638296887540, 1.23925878777028283880851799887, 2.75331774026099555259269055205, 3.70755338111861847890557482314, 4.58791687230002578278570098520, 5.45816243972202325474616209935, 6.63883509966765234219215139783, 7.20248586710915185967359257733, 8.528049585821896640918764093739, 9.612121079390389761005470693431, 10.01302268503376493112106692522

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