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Properties

Label 2-1045-5.4-c1-0-78
Degree $2$
Conductor $1045$
Sign $-0.954 + 0.299i$
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  − 1.23i·2-s − 2.07i·3-s + 0.469·4-s + (2.13 − 0.669i)5-s − 2.56·6-s − 0.314i·7-s − 3.05i·8-s − 1.29·9-s + (−0.827 − 2.63i)10-s + 11-s − 0.973i·12-s − 0.897i·13-s − 0.388·14-s + (−1.38 − 4.42i)15-s − 2.84·16-s + 0.623i·17-s + ⋯
L(s)  = 1  − 0.874i·2-s − 1.19i·3-s + 0.234·4-s + (0.954 − 0.299i)5-s − 1.04·6-s − 0.118i·7-s − 1.08i·8-s − 0.432·9-s + (−0.261 − 0.834i)10-s + 0.301·11-s − 0.280i·12-s − 0.248i·13-s − 0.103·14-s + (−0.358 − 1.14i)15-s − 0.710·16-s + 0.151i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.954 + 0.299i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.258591017\)
\(L(\frac12)\) \(\approx\) \(2.258591017\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.13 + 0.669i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + 1.23iT - 2T^{2} \)
3 \( 1 + 2.07iT - 3T^{2} \)
7 \( 1 + 0.314iT - 7T^{2} \)
13 \( 1 + 0.897iT - 13T^{2} \)
17 \( 1 - 0.623iT - 17T^{2} \)
23 \( 1 - 7.17iT - 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + 1.16T + 31T^{2} \)
37 \( 1 - 1.42iT - 37T^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 - 5.18iT - 43T^{2} \)
47 \( 1 + 4.33iT - 47T^{2} \)
53 \( 1 + 3.29iT - 53T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 + 5.74T + 61T^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 5.19T + 71T^{2} \)
73 \( 1 - 14.1iT - 73T^{2} \)
79 \( 1 + 4.00T + 79T^{2} \)
83 \( 1 - 1.65iT - 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 0.809iT - 97T^{2} \)
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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

−9.744023407243127776627363277077, −8.882823063790005576413396013039, −7.70886068298150562010523183509, −7.02368270802697017074851479164, −6.24200112366787426442682498316, −5.41009377629782986823464885631, −3.92379759419007197212414662015, −2.73044706317102426004624194425, −1.77213508630100390124912520507, −1.09288492504500504009547605867, 1.98452077734187674249273297612, 3.12016791349203581164087638786, 4.42589968635096780401209007517, 5.24739237329897518707318588245, 6.05023292052043865412770549150, 6.77403458885033006730763953027, 7.67063475974219572074200846600, 8.923691866393289156802372106445, 9.254804041790471849167692085488, 10.41203156611128965896050171294

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