Properties

Label 2-1815-1.1-c3-0-191
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $107.088$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s − 7·4-s + 5·5-s − 3·6-s + 24·7-s + 15·8-s + 9·9-s − 5·10-s − 21·12-s − 22·13-s − 24·14-s + 15·15-s + 41·16-s + 14·17-s − 9·18-s + 20·19-s − 35·20-s + 72·21-s − 168·23-s + 45·24-s + 25·25-s + 22·26-s + 27·27-s − 168·28-s − 230·29-s − 15·30-s + ⋯
L(s)  = 1  − 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s − 0.204·6-s + 1.29·7-s + 0.662·8-s + 1/3·9-s − 0.158·10-s − 0.505·12-s − 0.469·13-s − 0.458·14-s + 0.258·15-s + 0.640·16-s + 0.199·17-s − 0.117·18-s + 0.241·19-s − 0.391·20-s + 0.748·21-s − 1.52·23-s + 0.382·24-s + 1/5·25-s + 0.165·26-s + 0.192·27-s − 1.13·28-s − 1.47·29-s − 0.0912·30-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: yes
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 \( 1 - p T \)
5 \( 1 - p T \)
11 \( 1 \)
good2 \( 1 + T + p^{3} T^{2} \)
7 \( 1 - 24 T + p^{3} T^{2} \)
13 \( 1 + 22 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 + 168 T + p^{3} T^{2} \)
29 \( 1 + 230 T + p^{3} T^{2} \)
31 \( 1 + 288 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 + 122 T + p^{3} T^{2} \)
43 \( 1 - 188 T + p^{3} T^{2} \)
47 \( 1 - 256 T + p^{3} T^{2} \)
53 \( 1 + 338 T + p^{3} T^{2} \)
59 \( 1 - 100 T + p^{3} T^{2} \)
61 \( 1 + 742 T + p^{3} T^{2} \)
67 \( 1 + 84 T + p^{3} T^{2} \)
71 \( 1 + 328 T + p^{3} T^{2} \)
73 \( 1 - 38 T + p^{3} T^{2} \)
79 \( 1 - 240 T + p^{3} T^{2} \)
83 \( 1 + 1212 T + p^{3} T^{2} \)
89 \( 1 - 330 T + p^{3} T^{2} \)
97 \( 1 - 866 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.641569477463814017183032857649, −7.64238058691923613510999831675, −7.53014566471801599835571048915, −5.90313607328616586839250961595, −5.18409271346909968904575429061, −4.37656358417063700981146737998, −3.55507148039555621237041862497, −2.09321481864337913153636490983, −1.45391834572287050326864920532, 0, 1.45391834572287050326864920532, 2.09321481864337913153636490983, 3.55507148039555621237041862497, 4.37656358417063700981146737998, 5.18409271346909968904575429061, 5.90313607328616586839250961595, 7.53014566471801599835571048915, 7.64238058691923613510999831675, 8.641569477463814017183032857649

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