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Properties

Label 2-1520-5.4-c1-0-8
Degree $2$
Conductor $1520$
Sign $-0.316 - 0.948i$
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  + 2.41i·3-s + (−0.707 − 2.12i)5-s − 1.58i·7-s − 2.82·9-s − 1.41·11-s − 0.171i·13-s + (5.12 − 1.70i)15-s + i·17-s − 19-s + 3.82·21-s + 9.24i·23-s + (−3.99 + 3i)25-s + 0.414i·27-s + 5.82·29-s + 2.24·31-s + ⋯
L(s)  = 1  + 1.39i·3-s + (−0.316 − 0.948i)5-s − 0.599i·7-s − 0.942·9-s − 0.426·11-s − 0.0475i·13-s + (1.32 − 0.440i)15-s + 0.242i·17-s − 0.229·19-s + 0.835·21-s + 1.92i·23-s + (−0.799 + 0.600i)25-s + 0.0797i·27-s + 1.08·29-s + 0.402·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.235426437\)
\(L(\frac12)\) \(\approx\) \(1.235426437\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 2.12i)T \)
19 \( 1 + T \)
good3 \( 1 - 2.41iT - 3T^{2} \)
7 \( 1 + 1.58iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 0.171iT - 13T^{2} \)
17 \( 1 - iT - 17T^{2} \)
23 \( 1 - 9.24iT - 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 - 2.24T + 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 5.75T + 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 5.48iT - 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 2.48iT - 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−9.627132479003631601135618910052, −9.169047590365509353502440648024, −8.158456550909980280352510075621, −7.59937884328299366977492877694, −6.26713134262306995986363608635, −5.24308630160730822218457559836, −4.64259857386581524762749208680, −3.95450526932887332602264172729, −3.06253829825956769926206886081, −1.24957277443162886590991482531, 0.53058713385779782675361403418, 2.25388253093288475349575886833, 2.62845215827761559126014506746, 4.01013335217070721917736809449, 5.28535667006082211774502065414, 6.37148812261193787453585518580, 6.70060184641792138131439715967, 7.58578305248291793671381271608, 8.212807907123063006488923032748, 8.972397972489679383403637229854

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