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Properties

Label 2-1520-95.84-c0-0-0
Degree $2$
Conductor $1520$
Sign $-0.0977 - 0.995i$
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 − 11-s + (0.5 + 0.866i)19-s + (−0.499 − 0.866i)25-s + (−1.5 + 0.866i)29-s + 1.73i·31-s − 0.999·45-s + 49-s + (0.5 − 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 + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)19-s + (−0.499 − 0.866i)25-s + (−1.5 + 0.866i)29-s + 1.73i·31-s − 0.999·45-s + 49-s + (0.5 − 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 + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.0977 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8599742660\)
\(L(\frac12)\) \(\approx\) \(0.8599742660\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
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 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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

−10.23766869918517078886689117094, −9.042709256503115740392773612568, −8.038348030807971924120210666537, −7.50353021228434156258180763855, −6.87985390695981827174096518757, −5.69032167742024719550697154450, −4.95738910066557588974448966343, −3.82544330402758967059960250469, −2.95660145746773224237028649856, −1.83753685245591563050674928078, 0.68993234750026441642425620691, 2.26124630205559243726387100509, 3.58959396053843012967312407613, 4.35426245877518559416574605699, 5.28262147742168720290611435736, 6.07042886951349355883994356058, 7.30894181383640892232872846988, 7.73002991916927825318210164898, 8.737892582154781608433337579869, 9.407235513426562197219344897282

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