Properties

Label 2-1815-1.1-c1-0-32
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  + 0.262·2-s + 3-s − 1.93·4-s + 5-s + 0.262·6-s + 3.19·7-s − 1.03·8-s + 9-s + 0.262·10-s − 1.93·12-s + 1.11·13-s + 0.837·14-s + 15-s + 3.59·16-s + 0.0882·17-s + 0.262·18-s + 0.0688·19-s − 1.93·20-s + 3.19·21-s + 6.65·23-s − 1.03·24-s + 25-s + 0.293·26-s + 27-s − 6.16·28-s − 3.73·29-s + 0.262·30-s + ⋯
L(s)  = 1  + 0.185·2-s + 0.577·3-s − 0.965·4-s + 0.447·5-s + 0.107·6-s + 1.20·7-s − 0.364·8-s + 0.333·9-s + 0.0829·10-s − 0.557·12-s + 0.310·13-s + 0.223·14-s + 0.258·15-s + 0.897·16-s + 0.0213·17-s + 0.0618·18-s + 0.0157·19-s − 0.431·20-s + 0.696·21-s + 1.38·23-s − 0.210·24-s + 0.200·25-s + 0.0576·26-s + 0.192·27-s − 1.16·28-s − 0.693·29-s + 0.0479·30-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

\(L(1)\) \(\approx\) \(2.414968551\)
\(L(\frac12)\) \(\approx\) \(2.414968551\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.262T + 2T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
17 \( 1 - 0.0882T + 17T^{2} \)
19 \( 1 - 0.0688T + 19T^{2} \)
23 \( 1 - 6.65T + 23T^{2} \)
29 \( 1 + 3.73T + 29T^{2} \)
31 \( 1 + 9.58T + 31T^{2} \)
37 \( 1 + 8.33T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 0.908T + 47T^{2} \)
53 \( 1 - 0.872T + 53T^{2} \)
59 \( 1 - 1.83T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 - 9.53T + 67T^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 - 7.16T + 73T^{2} \)
79 \( 1 + 0.791T + 79T^{2} \)
83 \( 1 + 0.247T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 4.09T + 97T^{2} \)
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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

−9.033666768674567979715081616083, −8.721514399143117455706284799813, −7.78229257662763005447405710835, −7.10987667260612348340208346538, −5.72218829342170620948205678535, −5.19242919359963929494156329849, −4.29708468007252467834816122376, −3.49721483447658311495107856570, −2.24126647343059215216201602413, −1.10081133327573062986884208839, 1.10081133327573062986884208839, 2.24126647343059215216201602413, 3.49721483447658311495107856570, 4.29708468007252467834816122376, 5.19242919359963929494156329849, 5.72218829342170620948205678535, 7.10987667260612348340208346538, 7.78229257662763005447405710835, 8.721514399143117455706284799813, 9.033666768674567979715081616083

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