Properties

Label 2-1815-1.1-c3-0-119
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $107.088$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46·2-s + 3·3-s + 3.99·4-s − 5·5-s − 10.3·6-s − 27.7·7-s + 13.8·8-s + 9·9-s + 17.3·10-s + 11.9·12-s + 58.8·13-s + 95.9·14-s − 15·15-s − 80·16-s − 19.0·17-s − 31.1·18-s + 45.0·19-s − 19.9·20-s − 83.1·21-s − 75·23-s + 41.5·24-s + 25·25-s − 203.·26-s + 27·27-s − 110.·28-s + 128.·29-s + 51.9·30-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.447·5-s − 0.707·6-s − 1.49·7-s + 0.612·8-s + 0.333·9-s + 0.547·10-s + 0.288·12-s + 1.25·13-s + 1.83·14-s − 0.258·15-s − 1.25·16-s − 0.271·17-s − 0.408·18-s + 0.543·19-s − 0.223·20-s − 0.863·21-s − 0.679·23-s + 0.353·24-s + 0.200·25-s − 1.53·26-s + 0.192·27-s − 0.748·28-s + 0.820·29-s + 0.316·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(107.088\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1815,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
11 \( 1 \)
good2 \( 1 + 3.46T + 8T^{2} \)
7 \( 1 + 27.7T + 343T^{2} \)
13 \( 1 - 58.8T + 2.19e3T^{2} \)
17 \( 1 + 19.0T + 4.91e3T^{2} \)
19 \( 1 - 45.0T + 6.85e3T^{2} \)
23 \( 1 + 75T + 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 + 263T + 2.97e4T^{2} \)
37 \( 1 + 308T + 5.06e4T^{2} \)
41 \( 1 + 162.T + 6.89e4T^{2} \)
43 \( 1 - 38.1T + 7.95e4T^{2} \)
47 \( 1 - 93T + 1.03e5T^{2} \)
53 \( 1 - 525T + 1.48e5T^{2} \)
59 \( 1 - 498T + 2.05e5T^{2} \)
61 \( 1 - 441.T + 2.26e5T^{2} \)
67 \( 1 - 316T + 3.00e5T^{2} \)
71 \( 1 + 288T + 3.57e5T^{2} \)
73 \( 1 - 928.T + 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 571.T + 5.71e5T^{2} \)
89 \( 1 + 180T + 7.04e5T^{2} \)
97 \( 1 + 904T + 9.12e5T^{2} \)
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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

−8.701934162136508345247675401591, −7.972585256546923368305754714680, −7.06031645382877851742609598208, −6.56058481754417684266076064664, −5.37155222370367159050899813474, −3.93253661583584321483268229818, −3.48105588785586192220011544044, −2.22724285663359148253805578797, −1.00324702179408863786012740092, 0, 1.00324702179408863786012740092, 2.22724285663359148253805578797, 3.48105588785586192220011544044, 3.93253661583584321483268229818, 5.37155222370367159050899813474, 6.56058481754417684266076064664, 7.06031645382877851742609598208, 7.972585256546923368305754714680, 8.701934162136508345247675401591

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