Properties

Label 2-1815-1.1-c3-0-180
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s + 3·3-s − 0.701·4-s + 5·5-s − 8.10·6-s + 10.1·7-s + 23.5·8-s + 9·9-s − 13.5·10-s − 2.10·12-s + 26.7·13-s − 27.2·14-s + 15·15-s − 57.8·16-s − 36.1·17-s − 24.3·18-s − 66.3·19-s − 3.50·20-s + 30.3·21-s + 70.7·23-s + 70.5·24-s + 25·25-s − 72.1·26-s + 27·27-s − 7.08·28-s − 41.0·29-s − 40.5·30-s + ⋯
L(s)  = 1  − 0.955·2-s + 0.577·3-s − 0.0876·4-s + 0.447·5-s − 0.551·6-s + 0.545·7-s + 1.03·8-s + 0.333·9-s − 0.427·10-s − 0.0506·12-s + 0.569·13-s − 0.521·14-s + 0.258·15-s − 0.904·16-s − 0.516·17-s − 0.318·18-s − 0.800·19-s − 0.0392·20-s + 0.315·21-s + 0.641·23-s + 0.599·24-s + 0.200·25-s − 0.544·26-s + 0.192·27-s − 0.0478·28-s − 0.263·29-s − 0.246·30-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: \(1\)
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 \( 1 - 3T \)
5 \( 1 - 5T \)
11 \( 1 \)
good2 \( 1 + 2.70T + 8T^{2} \)
7 \( 1 - 10.1T + 343T^{2} \)
13 \( 1 - 26.7T + 2.19e3T^{2} \)
17 \( 1 + 36.1T + 4.91e3T^{2} \)
19 \( 1 + 66.3T + 6.85e3T^{2} \)
23 \( 1 - 70.7T + 1.21e4T^{2} \)
29 \( 1 + 41.0T + 2.43e4T^{2} \)
31 \( 1 + 24.2T + 2.97e4T^{2} \)
37 \( 1 + 367.T + 5.06e4T^{2} \)
41 \( 1 + 10.2T + 6.89e4T^{2} \)
43 \( 1 + 79.8T + 7.95e4T^{2} \)
47 \( 1 + 99.3T + 1.03e5T^{2} \)
53 \( 1 + 460.T + 1.48e5T^{2} \)
59 \( 1 + 753.T + 2.05e5T^{2} \)
61 \( 1 - 424.T + 2.26e5T^{2} \)
67 \( 1 + 1.04e3T + 3.00e5T^{2} \)
71 \( 1 - 263.T + 3.57e5T^{2} \)
73 \( 1 - 117.T + 3.89e5T^{2} \)
79 \( 1 - 707.T + 4.93e5T^{2} \)
83 \( 1 - 535.T + 5.71e5T^{2} \)
89 \( 1 + 823.T + 7.04e5T^{2} \)
97 \( 1 - 128.T + 9.12e5T^{2} \)
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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

−8.610531212247136331178139926527, −8.035122545384632581259806876668, −7.16812094674502770078198575038, −6.34489166767670248644275643056, −5.11954280602755234543018670329, −4.41224792007397389748013505921, −3.33119060220014748184995613980, −2.01314547814520529775825780244, −1.36718175397376415773211921976, 0, 1.36718175397376415773211921976, 2.01314547814520529775825780244, 3.33119060220014748184995613980, 4.41224792007397389748013505921, 5.11954280602755234543018670329, 6.34489166767670248644275643056, 7.16812094674502770078198575038, 8.035122545384632581259806876668, 8.610531212247136331178139926527

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