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Properties

Label 2-760-19.11-c1-0-15
Degree $2$
Conductor $760$
Sign $-0.128 + 0.991i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (1.37 − 2.38i)3-s + (−0.5 + 0.866i)5-s + 4.11·7-s + (−2.30 − 3.98i)9-s − 6.09·11-s + (−1.17 − 2.02i)13-s + (1.37 + 2.38i)15-s + (3.80 − 6.58i)17-s + (1.46 − 4.10i)19-s + (5.67 − 9.82i)21-s + (2.98 + 5.17i)23-s + (−0.499 − 0.866i)25-s − 4.43·27-s + (−0.969 − 1.67i)29-s − 1.43·31-s + ⋯
L(s)  = 1  + (0.796 − 1.37i)3-s + (−0.223 + 0.387i)5-s + 1.55·7-s + (−0.767 − 1.32i)9-s − 1.83·11-s + (−0.324 − 0.562i)13-s + (0.356 + 0.616i)15-s + (0.922 − 1.59i)17-s + (0.335 − 0.942i)19-s + (1.23 − 2.14i)21-s + (0.623 + 1.07i)23-s + (−0.0999 − 0.173i)25-s − 0.852·27-s + (−0.180 − 0.311i)29-s − 0.257·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ -0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32562 - 1.50880i\)
\(L(\frac12)\) \(\approx\) \(1.32562 - 1.50880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-1.46 + 4.10i)T \)
good3 \( 1 + (-1.37 + 2.38i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.11T + 7T^{2} \)
11 \( 1 + 6.09T + 11T^{2} \)
13 \( 1 + (1.17 + 2.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.80 + 6.58i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.98 - 5.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.969 + 1.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 - 8.91T + 37T^{2} \)
41 \( 1 + (1.70 - 2.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.655 + 1.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.192 + 0.333i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.597 + 1.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.48 - 9.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0825 + 0.142i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.848 - 1.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.75 - 8.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.97 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.87 + 6.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.25T + 83T^{2} \)
89 \( 1 + (-9.26 - 16.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.203 + 0.352i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.11102683205044496650409903198, −8.993141649118330382264987505815, −7.87214705095831362123478134021, −7.70820778875259348097385516266, −7.16302954998673308403312189918, −5.54322967018816639018518993393, −4.88179904783776517788191634722, −2.96585443405253634704878720392, −2.45274691473987810945664736513, −0.989747457857108387346557723790, 1.90062819870407155793676971019, 3.17824804643201823759064044138, 4.33129298366950069027021702796, 4.88881905298704823915797097157, 5.71787495341889889840442509833, 7.73051406978567686852542021363, 8.065529935753427234422589886699, 8.746629005761910544686760161350, 9.792323231873759782346267197922, 10.59363888396758172365149418650

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