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Properties

Label 2-1520-76.27-c1-0-23
Degree $2$
Conductor $1520$
Sign $0.991 - 0.133i$
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  + (−1.61 + 2.79i)3-s + (0.5 − 0.866i)5-s + 4.62i·7-s + (−3.69 − 6.39i)9-s − 5.58i·11-s + (−1.5 + 0.866i)13-s + (1.61 + 2.79i)15-s + (3.19 − 5.53i)17-s + (3.69 − 2.31i)19-s + (−12.9 − 7.45i)21-s + (3.53 − 2.04i)23-s + (−0.499 − 0.866i)25-s + 14.1·27-s + (4.82 − 2.78i)29-s − 3.53·31-s + ⋯
L(s)  = 1  + (−0.930 + 1.61i)3-s + (0.223 − 0.387i)5-s + 1.74i·7-s + (−1.23 − 2.13i)9-s − 1.68i·11-s + (−0.416 + 0.240i)13-s + (0.416 + 0.720i)15-s + (0.774 − 1.34i)17-s + (0.847 − 0.530i)19-s + (−2.81 − 1.62i)21-s + (0.738 − 0.426i)23-s + (−0.0999 − 0.173i)25-s + 2.72·27-s + (0.896 − 0.517i)29-s − 0.634·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.011325333\)
\(L(\frac12)\) \(\approx\) \(1.011325333\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.69 + 2.31i)T \)
good3 \( 1 + (1.61 - 2.79i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.62iT - 7T^{2} \)
11 \( 1 + 5.58iT - 11T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.19 + 5.53i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.53 + 2.04i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.82 + 2.78i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.53T + 31T^{2} \)
37 \( 1 + 2.10iT - 37T^{2} \)
41 \( 1 + (1.82 + 1.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.37 + 4.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.66 - 3.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.58 + 2.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.91 + 11.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.22 - 5.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.46 - 9.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.694 - 1.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.310 - 0.537i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.04iT - 83T^{2} \)
89 \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.326 - 0.188i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.470797220579203066921469942492, −8.946391613255255579583824945166, −8.352267259148091248264244233671, −6.70009979966328184827648888678, −5.76616586133887291388716951295, −5.29961101893140317600332775661, −4.86686742300486126178782510653, −3.42553263947422365448625269929, −2.76828100013969114227966470969, −0.52541677025316706727126369617, 1.14893321231402234824952829778, 1.78863819556918304799403318899, 3.31040544243743414432319708139, 4.62338952065154055085639945163, 5.45878362562295042732374877849, 6.52992022315847928431265269642, 7.05832742992327930178522052662, 7.52638076558565436220416040577, 8.138355943345659350142946456916, 9.833465362635179670778699176818

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