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Properties

Label 2-1520-380.159-c0-0-1
Degree $2$
Conductor $1520$
Sign $0.977 + 0.211i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)9-s − 1.73i·11-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)25-s + (0.5 − 0.866i)29-s + 1.73i·31-s + (1 + 1.73i)41-s + 0.999·45-s − 49-s + (1.49 − 0.866i)55-s + (−1.5 + 0.866i)59-s + (−0.5 + 0.866i)61-s + (1.5 − 0.866i)71-s + (−1.5 + 0.866i)79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)9-s − 1.73i·11-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)25-s + (0.5 − 0.866i)29-s + 1.73i·31-s + (1 + 1.73i)41-s + 0.999·45-s − 49-s + (1.49 − 0.866i)55-s + (−1.5 + 0.866i)59-s + (−0.5 + 0.866i)61-s + (1.5 − 0.866i)71-s + (−1.5 + 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.252422158\)
\(L(\frac12)\) \(\approx\) \(1.252422158\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.603719215726221964601517760798, −8.964108096063200209655686556520, −8.051979499326616417730632641467, −7.06007525636684195699814796283, −6.34914479170518942619213079663, −5.79716182483802234035147837302, −4.59157677232805852620433497212, −3.33741877174442598533052928101, −2.87179984536598896766796647761, −1.19479593689736668105221996145, 1.56852212430036046910997973783, 2.30532721374326435567173393619, 3.97860389539106826538131796932, 4.75491875274002610518996741225, 5.38045434340819595387331934285, 6.42957784686699256988661388612, 7.50009745542313518702915859435, 7.917319741949913610651196121430, 9.054368325783819712787400774422, 9.735047761983297755658855819874

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