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Properties

Label 2-1520-19.11-c1-0-11
Degree $2$
Conductor $1520$
Sign $0.141 - 0.989i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  + (−0.282 + 0.488i)3-s + (−0.5 + 0.866i)5-s + 1.56·7-s + (1.34 + 2.32i)9-s + 4.21·11-s + (−1.32 − 2.29i)13-s + (−0.282 − 0.488i)15-s + (−3.84 + 6.65i)17-s + (2.78 + 3.35i)19-s + (−0.441 + 0.764i)21-s + (−2.34 − 4.05i)23-s + (−0.499 − 0.866i)25-s − 3.20·27-s + (−1.60 − 2.78i)29-s + 8.81·31-s + ⋯
L(s)  = 1  + (−0.162 + 0.282i)3-s + (−0.223 + 0.387i)5-s + 0.591·7-s + (0.446 + 0.774i)9-s + 1.27·11-s + (−0.367 − 0.636i)13-s + (−0.0728 − 0.126i)15-s + (−0.931 + 1.61i)17-s + (0.638 + 0.769i)19-s + (−0.0962 + 0.166i)21-s + (−0.488 − 0.845i)23-s + (−0.0999 − 0.173i)25-s − 0.616·27-s + (−0.298 − 0.517i)29-s + 1.58·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.141 - 0.989i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.671065376\)
\(L(\frac12)\) \(\approx\) \(1.671065376\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.78 - 3.35i)T \)
good3 \( 1 + (0.282 - 0.488i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 - 4.21T + 11T^{2} \)
13 \( 1 + (1.32 + 2.29i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.84 - 6.65i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.34 + 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.60 + 2.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.81T + 31T^{2} \)
37 \( 1 - 3.56T + 37T^{2} \)
41 \( 1 + (1.21 - 2.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.23 + 3.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.17 + 2.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.57 - 4.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.607 + 1.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.57 - 11.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.24 - 7.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.01 - 10.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.19 - 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.56 - 6.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.37T + 83T^{2} \)
89 \( 1 + (7.63 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.71 - 8.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.09131150513572181745790217022, −8.631479698251718096680500807588, −8.161303875080705974887265104453, −7.27304856672892645026368258503, −6.36704007865659819411757946977, −5.58724936083130299398699219178, −4.34451433727467846992311334527, −4.02408375917547168583651730636, −2.52665916308854198567922636455, −1.40612414080863494647431196220, 0.74607714416742799479961913656, 1.83762390044130515244781233912, 3.26709701895622672715526805313, 4.40716242710355047758131745327, 4.88910565470643002813053012471, 6.19981982679530896904533962423, 6.91863769347705661040101239614, 7.49936718391349383105479611388, 8.608336066292474407996404772306, 9.458418052719167061713935417799

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