Properties

Label 2-1815-1.1-c3-0-141
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  − 3.59·2-s − 3·3-s + 4.89·4-s − 5·5-s + 10.7·6-s + 16.1·7-s + 11.1·8-s + 9·9-s + 17.9·10-s − 14.6·12-s + 54.1·13-s − 57.9·14-s + 15·15-s − 79.2·16-s + 107.·17-s − 32.3·18-s − 48.7·19-s − 24.4·20-s − 48.4·21-s + 11.9·23-s − 33.4·24-s + 25·25-s − 194.·26-s − 27·27-s + 78.9·28-s − 239.·29-s − 53.8·30-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.577·3-s + 0.611·4-s − 0.447·5-s + 0.732·6-s + 0.871·7-s + 0.493·8-s + 0.333·9-s + 0.567·10-s − 0.353·12-s + 1.15·13-s − 1.10·14-s + 0.258·15-s − 1.23·16-s + 1.52·17-s − 0.423·18-s − 0.588·19-s − 0.273·20-s − 0.503·21-s + 0.108·23-s − 0.284·24-s + 0.200·25-s − 1.46·26-s − 0.192·27-s + 0.533·28-s − 1.53·29-s − 0.327·30-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: Trivial
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 \( 1 + 3.59T + 8T^{2} \)
7 \( 1 - 16.1T + 343T^{2} \)
13 \( 1 - 54.1T + 2.19e3T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 + 48.7T + 6.85e3T^{2} \)
23 \( 1 - 11.9T + 1.21e4T^{2} \)
29 \( 1 + 239.T + 2.43e4T^{2} \)
31 \( 1 + 82.0T + 2.97e4T^{2} \)
37 \( 1 + 21.7T + 5.06e4T^{2} \)
41 \( 1 - 124.T + 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 + 186.T + 1.03e5T^{2} \)
53 \( 1 - 233.T + 1.48e5T^{2} \)
59 \( 1 - 232.T + 2.05e5T^{2} \)
61 \( 1 + 163.T + 2.26e5T^{2} \)
67 \( 1 + 876.T + 3.00e5T^{2} \)
71 \( 1 + 733.T + 3.57e5T^{2} \)
73 \( 1 + 1.16e3T + 3.89e5T^{2} \)
79 \( 1 - 588.T + 4.93e5T^{2} \)
83 \( 1 - 1.16e3T + 5.71e5T^{2} \)
89 \( 1 + 1.04e3T + 7.04e5T^{2} \)
97 \( 1 - 1.54e3T + 9.12e5T^{2} \)
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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

−8.547752296022078209545184844044, −7.76481259739063057927867041688, −7.35863018972085559916540332048, −6.21092342641566896105627523170, −5.34509162263331408617604789477, −4.40638030267282377004497718832, −3.47423310069009894467754529805, −1.78205462058211563833813816906, −1.11443549351482892981802612504, 0, 1.11443549351482892981802612504, 1.78205462058211563833813816906, 3.47423310069009894467754529805, 4.40638030267282377004497718832, 5.34509162263331408617604789477, 6.21092342641566896105627523170, 7.35863018972085559916540332048, 7.76481259739063057927867041688, 8.547752296022078209545184844044

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