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Properties

Label 2-1519-1519.1332-c0-0-0
Degree $2$
Conductor $1519$
Sign $0.989 + 0.141i$
Analytic cond. $0.758079$
Root an. cond. $0.870677$
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  + (−1.28 − 1.19i)2-s + (0.155 + 2.07i)4-s + (0.727 + 1.85i)5-s + (0.995 − 0.0995i)7-s + (1.18 − 1.49i)8-s + (0.955 + 0.294i)9-s + (1.27 − 3.25i)10-s + (−1.40 − 1.06i)14-s + (−1.24 + 0.188i)16-s + (−0.878 − 1.52i)18-s + (0.797 − 1.38i)19-s + (−3.73 + 1.80i)20-s + (−2.17 + 2.01i)25-s + (0.362 + 2.05i)28-s + (−0.5 − 0.866i)31-s + (0.257 + 0.175i)32-s + ⋯
L(s)  = 1  + (−1.28 − 1.19i)2-s + (0.155 + 2.07i)4-s + (0.727 + 1.85i)5-s + (0.995 − 0.0995i)7-s + (1.18 − 1.49i)8-s + (0.955 + 0.294i)9-s + (1.27 − 3.25i)10-s + (−1.40 − 1.06i)14-s + (−1.24 + 0.188i)16-s + (−0.878 − 1.52i)18-s + (0.797 − 1.38i)19-s + (−3.73 + 1.80i)20-s + (−2.17 + 2.01i)25-s + (0.362 + 2.05i)28-s + (−0.5 − 0.866i)31-s + (0.257 + 0.175i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (1332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ 0.989 + 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7937500251\)
\(L(\frac12)\) \(\approx\) \(0.7937500251\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.995 + 0.0995i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.28 + 1.19i)T + (0.0747 + 0.997i)T^{2} \)
3 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.727 - 1.85i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (-0.797 + 1.38i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (-0.337 + 0.423i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.729 - 1.85i)T + (-0.733 - 0.680i)T^{2} \)
61 \( 1 + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.0449 - 0.0216i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + 1.93T + T^{2} \)
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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

−9.702843251356892809595967810475, −9.300901407866118782846460558064, −8.082757925208247084445045838236, −7.32753633681579398837472405390, −6.95881509326501967763130315658, −5.59796548072826417814178129800, −4.22306578892537663226830505252, −3.09893955499845876166722047083, −2.33794740429051703376720991787, −1.53108611125116879725503681653, 1.24844463460496747016585554932, 1.62505376519468124705875339622, 4.18699211677825103514071601332, 5.13071439282106180396690933353, 5.59390877693395550924985798059, 6.53143169392901809085265402620, 7.64243773873429476382381169251, 8.124920734582433371204799100805, 8.771688315841014739967065918590, 9.542363021745650996743961097995

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