Properties

Label 2-1815-1.1-c3-0-206
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

−8.481055845634871242017171902337, −7.57737050200561367640296464145, −7.07385951830119899125341657325, −5.69466398987297751106240968308, −5.10526581403036087590839088111, −4.31552042648913352660920212143, −3.73139154377647754815279286811, −2.56593384136032162932678342029, −1.75485770745531347933769299760, 0, 1.75485770745531347933769299760, 2.56593384136032162932678342029, 3.73139154377647754815279286811, 4.31552042648913352660920212143, 5.10526581403036087590839088111, 5.69466398987297751106240968308, 7.07385951830119899125341657325, 7.57737050200561367640296464145, 8.481055845634871242017171902337

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