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Properties

Label 2-1045-209.208-c1-0-36
Degree $2$
Conductor $1045$
Sign $0.418 - 0.908i$
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  − 0.233·2-s + 2.00i·3-s − 1.94·4-s + 5-s − 0.467i·6-s + 1.43i·7-s + 0.921·8-s − 1.00·9-s − 0.233·10-s + (2.74 − 1.86i)11-s − 3.89i·12-s + 5.35·13-s − 0.334i·14-s + 2.00i·15-s + 3.67·16-s − 4.91i·17-s + ⋯
L(s)  = 1  − 0.165·2-s + 1.15i·3-s − 0.972·4-s + 0.447·5-s − 0.190i·6-s + 0.541i·7-s + 0.325·8-s − 0.335·9-s − 0.0738·10-s + (0.826 − 0.562i)11-s − 1.12i·12-s + 1.48·13-s − 0.0894i·14-s + 0.516i·15-s + 0.918·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.418 - 0.908i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.497253018\)
\(L(\frac12)\) \(\approx\) \(1.497253018\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.74 + 1.86i)T \)
19 \( 1 + (-3.73 + 2.24i)T \)
good2 \( 1 + 0.233T + 2T^{2} \)
3 \( 1 - 2.00iT - 3T^{2} \)
7 \( 1 - 1.43iT - 7T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 + 4.91iT - 17T^{2} \)
23 \( 1 + 4.98T + 23T^{2} \)
29 \( 1 - 4.09T + 29T^{2} \)
31 \( 1 - 3.87iT - 31T^{2} \)
37 \( 1 + 5.67iT - 37T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 - 9.77iT - 43T^{2} \)
47 \( 1 + 9.64T + 47T^{2} \)
53 \( 1 - 6.67iT - 53T^{2} \)
59 \( 1 - 3.93iT - 59T^{2} \)
61 \( 1 + 9.62iT - 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 + 1.57iT - 71T^{2} \)
73 \( 1 - 9.80iT - 73T^{2} \)
79 \( 1 + 1.60T + 79T^{2} \)
83 \( 1 + 8.47iT - 83T^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 - 13.9iT - 97T^{2} \)
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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

−9.791031547796624738536521095316, −9.253512675930649897745280066011, −8.858022845126801052863358837451, −7.88656794886049346943813436828, −6.46020357109211848149072973555, −5.59230123243245700473012067783, −4.80500259748655005965313873746, −3.93146788686998784522040001435, −3.09272223958247475420959429719, −1.13911708954509667368378394277, 1.04206389259013248687505775196, 1.75363450591584601592192470285, 3.64628271819346926463648837045, 4.32337955986177717296064528958, 5.77083382786722338100814437251, 6.36349291037624509926103337780, 7.32478479832514912424895740092, 8.191692857100776765906814117376, 8.722964418799955623253833495482, 9.882421235169677521723302227179

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