Properties

Label 2-1815-1.1-c1-0-42
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.45·2-s + 3-s + 4.03·4-s − 5-s − 2.45·6-s − 3.28·7-s − 4.98·8-s + 9-s + 2.45·10-s + 4.03·12-s + 0.313·13-s + 8.07·14-s − 15-s + 4.18·16-s − 5·17-s − 2.45·18-s + 7.45·19-s − 4.03·20-s − 3.28·21-s + 1.07·23-s − 4.98·24-s + 25-s − 0.769·26-s + 27-s − 13.2·28-s + 5.03·29-s + 2.45·30-s + ⋯
L(s)  = 1  − 1.73·2-s + 0.577·3-s + 2.01·4-s − 0.447·5-s − 1.00·6-s − 1.24·7-s − 1.76·8-s + 0.333·9-s + 0.776·10-s + 1.16·12-s + 0.0868·13-s + 2.15·14-s − 0.258·15-s + 1.04·16-s − 1.21·17-s − 0.578·18-s + 1.71·19-s − 0.901·20-s − 0.717·21-s + 0.223·23-s − 1.01·24-s + 0.200·25-s − 0.150·26-s + 0.192·27-s − 2.50·28-s + 0.935·29-s + 0.448·30-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 \( 1 + 2.45T + 2T^{2} \)
7 \( 1 + 3.28T + 7T^{2} \)
13 \( 1 - 0.313T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 7.45T + 19T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 - 3.44T + 31T^{2} \)
37 \( 1 - 2.63T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 4.93T + 53T^{2} \)
59 \( 1 + 9.16T + 59T^{2} \)
61 \( 1 + 9.18T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 - 3.07T + 71T^{2} \)
73 \( 1 - 8.65T + 73T^{2} \)
79 \( 1 + 5.41T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 - 0.224T + 97T^{2} \)
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

−8.908667234655734692486962823402, −8.290501363977722618610176430068, −7.46861972435229994282584674935, −6.85380926165783537258127240312, −6.19086467283757828296315888362, −4.66484749668547973980680370742, −3.31114649945115415275078150721, −2.70371630955836292923015196050, −1.32401833821696226416551924500, 0, 1.32401833821696226416551924500, 2.70371630955836292923015196050, 3.31114649945115415275078150721, 4.66484749668547973980680370742, 6.19086467283757828296315888362, 6.85380926165783537258127240312, 7.46861972435229994282584674935, 8.290501363977722618610176430068, 8.908667234655734692486962823402

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