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Properties

Label 2-1045-5.4-c1-0-90
Degree $2$
Conductor $1045$
Sign $0.847 - 0.531i$
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  − 2.74i·2-s − 2.61i·3-s − 5.52·4-s + (1.89 − 1.18i)5-s − 7.17·6-s − 3.68i·7-s + 9.67i·8-s − 3.83·9-s + (−3.26 − 5.19i)10-s − 11-s + 14.4i·12-s + 5.13i·13-s − 10.1·14-s + (−3.10 − 4.95i)15-s + 15.5·16-s − 1.38i·17-s + ⋯
L(s)  = 1  − 1.94i·2-s − 1.50i·3-s − 2.76·4-s + (0.847 − 0.531i)5-s − 2.92·6-s − 1.39i·7-s + 3.42i·8-s − 1.27·9-s + (−1.03 − 1.64i)10-s − 0.301·11-s + 4.17i·12-s + 1.42i·13-s − 2.70·14-s + (−0.802 − 1.27i)15-s + 3.87·16-s − 0.334i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.847 - 0.531i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.847 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.091481500\)
\(L(\frac12)\) \(\approx\) \(1.091481500\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.89 + 1.18i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + 2.74iT - 2T^{2} \)
3 \( 1 + 2.61iT - 3T^{2} \)
7 \( 1 + 3.68iT - 7T^{2} \)
13 \( 1 - 5.13iT - 13T^{2} \)
17 \( 1 + 1.38iT - 17T^{2} \)
23 \( 1 - 2.22iT - 23T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 + 6.00T + 31T^{2} \)
37 \( 1 - 10.0iT - 37T^{2} \)
41 \( 1 - 1.40T + 41T^{2} \)
43 \( 1 + 9.33iT - 43T^{2} \)
47 \( 1 + 4.35iT - 47T^{2} \)
53 \( 1 + 7.73iT - 53T^{2} \)
59 \( 1 - 0.926T + 59T^{2} \)
61 \( 1 - 0.458T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 - 7.40T + 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 2.23iT - 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + 1.79iT - 97T^{2} \)
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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

−9.410799895366478487591043457257, −8.622085025515950326942309456571, −7.62393978007727100296769409486, −6.74401580867013254052658280775, −5.45010298008036029685414707103, −4.47756975640682946042112714517, −3.44142850902764168956812574025, −2.03474570687975808227942792885, −1.59716454926455168576526620840, −0.50709202977781330371984977459, 2.87837578013343703883151605673, 3.97907276225792915267590215205, 5.19211949086056053414370155612, 5.58566757290804543118922148747, 6.03959806043164114519080107653, 7.32903560707117868353574944244, 8.234545240416898102673879956138, 9.118997974181334019063129324004, 9.423533855513174281446298003624, 10.25899018536747927630021021050

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