Properties

Label 2-1815-1.1-c1-0-60
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.16·2-s + 3-s − 0.649·4-s − 5-s − 1.16·6-s + 4.28·7-s + 3.07·8-s + 9-s + 1.16·10-s − 0.649·12-s − 5.16·13-s − 4.98·14-s − 15-s − 2.27·16-s − 5·17-s − 1.16·18-s − 5.59·19-s + 0.649·20-s + 4.28·21-s − 0.219·23-s + 3.07·24-s + 25-s + 6.00·26-s + 27-s − 2.78·28-s − 6.41·29-s + 1.16·30-s + ⋯
L(s)  = 1  − 0.821·2-s + 0.577·3-s − 0.324·4-s − 0.447·5-s − 0.474·6-s + 1.62·7-s + 1.08·8-s + 0.333·9-s + 0.367·10-s − 0.187·12-s − 1.43·13-s − 1.33·14-s − 0.258·15-s − 0.569·16-s − 1.21·17-s − 0.273·18-s − 1.28·19-s + 0.145·20-s + 0.935·21-s − 0.0458·23-s + 0.628·24-s + 0.200·25-s + 1.17·26-s + 0.192·27-s − 0.526·28-s − 1.19·29-s + 0.212·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 1.16T + 2T^{2} \)
7 \( 1 - 4.28T + 7T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 5.59T + 19T^{2} \)
23 \( 1 + 0.219T + 23T^{2} \)
29 \( 1 + 6.41T + 29T^{2} \)
31 \( 1 + 2.83T + 31T^{2} \)
37 \( 1 - 3.92T + 37T^{2} \)
41 \( 1 + 5.86T + 41T^{2} \)
43 \( 1 - 8.90T + 43T^{2} \)
47 \( 1 + 0.237T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 + 7.87T + 59T^{2} \)
61 \( 1 - 4.85T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 9.98T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 8.00T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 2.56T + 89T^{2} \)
97 \( 1 - 2.01T + 97T^{2} \)
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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

−8.780192512574325990719240719714, −8.210787137813819728752829612727, −7.56531313340157296863428848399, −7.00573758078770365782765787161, −5.42137173514394006587323050218, −4.46610126456342554425572227736, −4.20214309889305666672614843019, −2.42438114608179200841232967448, −1.64996811205198278011163227357, 0, 1.64996811205198278011163227357, 2.42438114608179200841232967448, 4.20214309889305666672614843019, 4.46610126456342554425572227736, 5.42137173514394006587323050218, 7.00573758078770365782765787161, 7.56531313340157296863428848399, 8.210787137813819728752829612727, 8.780192512574325990719240719714

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