Properties

Label 2-15-1.1-c11-0-0
Degree $2$
Conductor $15$
Sign $1$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27.7·2-s − 243·3-s − 1.27e3·4-s − 3.12e3·5-s + 6.73e3·6-s − 7.21e4·7-s + 9.22e4·8-s + 5.90e4·9-s + 8.66e4·10-s − 5.09e5·11-s + 3.10e5·12-s + 1.85e6·13-s + 2.00e6·14-s + 7.59e5·15-s + 6.39e4·16-s + 5.94e6·17-s − 1.63e6·18-s + 6.02e6·19-s + 3.99e6·20-s + 1.75e7·21-s + 1.41e7·22-s − 4.82e7·23-s − 2.24e7·24-s + 9.76e6·25-s − 5.14e7·26-s − 1.43e7·27-s + 9.23e7·28-s + ⋯
L(s)  = 1  − 0.612·2-s − 0.577·3-s − 0.624·4-s − 0.447·5-s + 0.353·6-s − 1.62·7-s + 0.995·8-s + 0.333·9-s + 0.273·10-s − 0.953·11-s + 0.360·12-s + 1.38·13-s + 0.993·14-s + 0.258·15-s + 0.0152·16-s + 1.01·17-s − 0.204·18-s + 0.557·19-s + 0.279·20-s + 0.936·21-s + 0.584·22-s − 1.56·23-s − 0.574·24-s + 0.199·25-s − 0.848·26-s − 0.192·27-s + 1.01·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $1$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.4802770143\)
\(L(\frac12)\) \(\approx\) \(0.4802770143\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
5 \( 1 + 3.12e3T \)
good2 \( 1 + 27.7T + 2.04e3T^{2} \)
7 \( 1 + 7.21e4T + 1.97e9T^{2} \)
11 \( 1 + 5.09e5T + 2.85e11T^{2} \)
13 \( 1 - 1.85e6T + 1.79e12T^{2} \)
17 \( 1 - 5.94e6T + 3.42e13T^{2} \)
19 \( 1 - 6.02e6T + 1.16e14T^{2} \)
23 \( 1 + 4.82e7T + 9.52e14T^{2} \)
29 \( 1 + 1.13e7T + 1.22e16T^{2} \)
31 \( 1 + 1.72e8T + 2.54e16T^{2} \)
37 \( 1 - 6.25e8T + 1.77e17T^{2} \)
41 \( 1 + 5.53e8T + 5.50e17T^{2} \)
43 \( 1 - 1.52e9T + 9.29e17T^{2} \)
47 \( 1 - 1.19e9T + 2.47e18T^{2} \)
53 \( 1 - 1.22e9T + 9.26e18T^{2} \)
59 \( 1 + 5.83e9T + 3.01e19T^{2} \)
61 \( 1 + 6.61e9T + 4.35e19T^{2} \)
67 \( 1 - 1.66e10T + 1.22e20T^{2} \)
71 \( 1 - 7.36e9T + 2.31e20T^{2} \)
73 \( 1 + 6.35e9T + 3.13e20T^{2} \)
79 \( 1 - 2.47e10T + 7.47e20T^{2} \)
83 \( 1 - 3.59e10T + 1.28e21T^{2} \)
89 \( 1 - 7.47e10T + 2.77e21T^{2} \)
97 \( 1 + 1.66e10T + 7.15e21T^{2} \)
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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

−16.50717456802371117733134398414, −15.89747561368340138259411776309, −13.60801005543520109249561456787, −12.52040080732156116522683679583, −10.61436274528338911663302044329, −9.476870678758022322826466147871, −7.77524513218051268282626188862, −5.85383030287690847270410174135, −3.70619148320679225681175072859, −0.60381703179539193221583408932, 0.60381703179539193221583408932, 3.70619148320679225681175072859, 5.85383030287690847270410174135, 7.77524513218051268282626188862, 9.476870678758022322826466147871, 10.61436274528338911663302044329, 12.52040080732156116522683679583, 13.60801005543520109249561456787, 15.89747561368340138259411776309, 16.50717456802371117733134398414

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