Properties

Label 2-15-1.1-c11-0-1
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  − 74.9·2-s + 243·3-s + 3.57e3·4-s + 3.12e3·5-s − 1.82e4·6-s − 6.30e4·7-s − 1.14e5·8-s + 5.90e4·9-s − 2.34e5·10-s + 2.56e5·11-s + 8.68e5·12-s − 4.65e5·13-s + 4.72e6·14-s + 7.59e5·15-s + 1.25e6·16-s + 1.08e7·17-s − 4.42e6·18-s + 9.38e6·19-s + 1.11e7·20-s − 1.53e7·21-s − 1.92e7·22-s − 2.41e7·23-s − 2.77e7·24-s + 9.76e6·25-s + 3.49e7·26-s + 1.43e7·27-s − 2.25e8·28-s + ⋯
L(s)  = 1  − 1.65·2-s + 0.577·3-s + 1.74·4-s + 0.447·5-s − 0.956·6-s − 1.41·7-s − 1.23·8-s + 0.333·9-s − 0.740·10-s + 0.480·11-s + 1.00·12-s − 0.347·13-s + 2.34·14-s + 0.258·15-s + 0.299·16-s + 1.85·17-s − 0.552·18-s + 0.869·19-s + 0.780·20-s − 0.818·21-s − 0.795·22-s − 0.781·23-s − 0.712·24-s + 0.199·25-s + 0.576·26-s + 0.192·27-s − 2.47·28-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

\(L(6)\) \(\approx\) \(0.9461668019\)
\(L(\frac12)\) \(\approx\) \(0.9461668019\)
\(L(\frac{13}{2})\) 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 \( 1 + 74.9T + 2.04e3T^{2} \)
7 \( 1 + 6.30e4T + 1.97e9T^{2} \)
11 \( 1 - 2.56e5T + 2.85e11T^{2} \)
13 \( 1 + 4.65e5T + 1.79e12T^{2} \)
17 \( 1 - 1.08e7T + 3.42e13T^{2} \)
19 \( 1 - 9.38e6T + 1.16e14T^{2} \)
23 \( 1 + 2.41e7T + 9.52e14T^{2} \)
29 \( 1 - 2.07e8T + 1.22e16T^{2} \)
31 \( 1 - 2.06e8T + 2.54e16T^{2} \)
37 \( 1 - 2.18e8T + 1.77e17T^{2} \)
41 \( 1 - 8.54e8T + 5.50e17T^{2} \)
43 \( 1 + 2.70e8T + 9.29e17T^{2} \)
47 \( 1 + 2.09e9T + 2.47e18T^{2} \)
53 \( 1 + 1.20e9T + 9.26e18T^{2} \)
59 \( 1 - 6.04e9T + 3.01e19T^{2} \)
61 \( 1 + 1.28e10T + 4.35e19T^{2} \)
67 \( 1 - 6.01e9T + 1.22e20T^{2} \)
71 \( 1 - 7.99e9T + 2.31e20T^{2} \)
73 \( 1 - 3.23e10T + 3.13e20T^{2} \)
79 \( 1 - 3.84e9T + 7.47e20T^{2} \)
83 \( 1 + 1.29e9T + 1.28e21T^{2} \)
89 \( 1 + 6.96e10T + 2.77e21T^{2} \)
97 \( 1 - 2.33e10T + 7.15e21T^{2} \)
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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

−16.73750429946503667993551495141, −15.87906609781069575280976312103, −14.01721976863694618361139233977, −12.20243374244652704098083399569, −10.00772480036113087759238202734, −9.592079496797552020896706665393, −7.995520899264134392699618666307, −6.49390758188442883354091661666, −2.92046330832211970092033260221, −0.981473718487176865459024094512, 0.981473718487176865459024094512, 2.92046330832211970092033260221, 6.49390758188442883354091661666, 7.995520899264134392699618666307, 9.592079496797552020896706665393, 10.00772480036113087759238202734, 12.20243374244652704098083399569, 14.01721976863694618361139233977, 15.87906609781069575280976312103, 16.73750429946503667993551495141

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