Properties

Label 2-1815-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 3-s − 1.61·4-s − 5-s + 0.618·6-s + 3.23·7-s + 2.23·8-s + 9-s + 0.618·10-s + 1.61·12-s + 5.23·13-s − 2.00·14-s + 15-s + 1.85·16-s − 5.47·17-s − 0.618·18-s + 6.47·19-s + 1.61·20-s − 3.23·21-s − 4.70·23-s − 2.23·24-s + 25-s − 3.23·26-s − 27-s − 5.23·28-s − 1.23·29-s − 0.618·30-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.447·5-s + 0.252·6-s + 1.22·7-s + 0.790·8-s + 0.333·9-s + 0.195·10-s + 0.467·12-s + 1.45·13-s − 0.534·14-s + 0.258·15-s + 0.463·16-s − 1.32·17-s − 0.145·18-s + 1.48·19-s + 0.361·20-s − 0.706·21-s − 0.981·23-s − 0.456·24-s + 0.200·25-s − 0.634·26-s − 0.192·27-s − 0.989·28-s − 0.229·29-s − 0.112·30-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: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9736861355\)
\(L(\frac12)\) \(\approx\) \(0.9736861355\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
7 \( 1 - 3.23T + 7T^{2} \)
13 \( 1 - 5.23T + 13T^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + 1.23T + 29T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 0.763T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 - 8.70T + 47T^{2} \)
53 \( 1 - 9.94T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 1.47T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−9.037052855310290616352642500388, −8.582099751421481240773285062363, −7.79315117774579275858784669052, −7.13298187166834420041071977118, −5.87906373830787910810902376787, −5.19520931276398807688658271499, −4.30874286982828647610998319217, −3.70187628979622433491499798442, −1.83600078314410590847902660307, −0.78651131815844179878191431538, 0.78651131815844179878191431538, 1.83600078314410590847902660307, 3.70187628979622433491499798442, 4.30874286982828647610998319217, 5.19520931276398807688658271499, 5.87906373830787910810902376787, 7.13298187166834420041071977118, 7.79315117774579275858784669052, 8.582099751421481240773285062363, 9.037052855310290616352642500388

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