Properties

Label 2-1815-1.1-c3-0-79
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s + 3·3-s − 5.56·4-s − 5·5-s − 4.68·6-s + 10.2·7-s + 21.1·8-s + 9·9-s + 7.80·10-s − 16.6·12-s + 40.8·13-s − 16·14-s − 15·15-s + 11.4·16-s + 98.7·17-s − 14.0·18-s + 39.6·19-s + 27.8·20-s + 30.7·21-s + 61.6·23-s + 63.5·24-s + 25·25-s − 63.8·26-s + 27·27-s − 56.9·28-s + 149.·29-s + 23.4·30-s + ⋯
L(s)  = 1  − 0.552·2-s + 0.577·3-s − 0.695·4-s − 0.447·5-s − 0.318·6-s + 0.553·7-s + 0.935·8-s + 0.333·9-s + 0.246·10-s − 0.401·12-s + 0.872·13-s − 0.305·14-s − 0.258·15-s + 0.178·16-s + 1.40·17-s − 0.184·18-s + 0.478·19-s + 0.310·20-s + 0.319·21-s + 0.559·23-s + 0.540·24-s + 0.200·25-s − 0.481·26-s + 0.192·27-s − 0.384·28-s + 0.954·29-s + 0.142·30-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: \(0\)
Selberg data: \((2,\ 1815,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.003615445\)
\(L(\frac12)\) \(\approx\) \(2.003615445\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.56T + 8T^{2} \)
7 \( 1 - 10.2T + 343T^{2} \)
13 \( 1 - 40.8T + 2.19e3T^{2} \)
17 \( 1 - 98.7T + 4.91e3T^{2} \)
19 \( 1 - 39.6T + 6.85e3T^{2} \)
23 \( 1 - 61.6T + 1.21e4T^{2} \)
29 \( 1 - 149.T + 2.43e4T^{2} \)
31 \( 1 - 54.7T + 2.97e4T^{2} \)
37 \( 1 - 44.8T + 5.06e4T^{2} \)
41 \( 1 + 336.T + 6.89e4T^{2} \)
43 \( 1 - 2.36T + 7.95e4T^{2} \)
47 \( 1 + 333.T + 1.03e5T^{2} \)
53 \( 1 - 640.T + 1.48e5T^{2} \)
59 \( 1 + 370.T + 2.05e5T^{2} \)
61 \( 1 - 714.T + 2.26e5T^{2} \)
67 \( 1 + 404.T + 3.00e5T^{2} \)
71 \( 1 - 939.T + 3.57e5T^{2} \)
73 \( 1 - 362.T + 3.89e5T^{2} \)
79 \( 1 + 951.T + 4.93e5T^{2} \)
83 \( 1 + 735.T + 5.71e5T^{2} \)
89 \( 1 - 385.T + 7.04e5T^{2} \)
97 \( 1 + 966.T + 9.12e5T^{2} \)
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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

−8.626895943751117484650205134830, −8.329416802262843097606747714239, −7.66648913910745085004576431271, −6.80865577436549044515143411398, −5.50761729835406809003913038289, −4.75949872444302853093417180656, −3.82699095849970679249022270877, −3.07576153527720861045827526549, −1.52701016907896674242472741761, −0.790390644580710781391790215407, 0.790390644580710781391790215407, 1.52701016907896674242472741761, 3.07576153527720861045827526549, 3.82699095849970679249022270877, 4.75949872444302853093417180656, 5.50761729835406809003913038289, 6.80865577436549044515143411398, 7.66648913910745085004576431271, 8.329416802262843097606747714239, 8.626895943751117484650205134830

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