Properties

Label 2-1815-1.1-c3-0-136
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  − 5.43·2-s + 3·3-s + 21.5·4-s − 5·5-s − 16.3·6-s + 9.68·7-s − 73.5·8-s + 9·9-s + 27.1·10-s + 64.6·12-s − 91.7·13-s − 52.6·14-s − 15·15-s + 227.·16-s − 65.9·17-s − 48.9·18-s + 118.·19-s − 107.·20-s + 29.0·21-s − 113.·23-s − 220.·24-s + 25·25-s + 498.·26-s + 27·27-s + 208.·28-s − 54.6·29-s + 81.5·30-s + ⋯
L(s)  = 1  − 1.92·2-s + 0.577·3-s + 2.69·4-s − 0.447·5-s − 1.10·6-s + 0.522·7-s − 3.25·8-s + 0.333·9-s + 0.859·10-s + 1.55·12-s − 1.95·13-s − 1.00·14-s − 0.258·15-s + 3.55·16-s − 0.940·17-s − 0.640·18-s + 1.43·19-s − 1.20·20-s + 0.301·21-s − 1.03·23-s − 1.87·24-s + 0.200·25-s + 3.76·26-s + 0.192·27-s + 1.40·28-s − 0.349·29-s + 0.496·30-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: Trivial
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 \( 1 + 5.43T + 8T^{2} \)
7 \( 1 - 9.68T + 343T^{2} \)
13 \( 1 + 91.7T + 2.19e3T^{2} \)
17 \( 1 + 65.9T + 4.91e3T^{2} \)
19 \( 1 - 118.T + 6.85e3T^{2} \)
23 \( 1 + 113.T + 1.21e4T^{2} \)
29 \( 1 + 54.6T + 2.43e4T^{2} \)
31 \( 1 - 288.T + 2.97e4T^{2} \)
37 \( 1 - 123.T + 5.06e4T^{2} \)
41 \( 1 - 209.T + 6.89e4T^{2} \)
43 \( 1 - 9.24T + 7.95e4T^{2} \)
47 \( 1 - 282.T + 1.03e5T^{2} \)
53 \( 1 - 352.T + 1.48e5T^{2} \)
59 \( 1 - 594.T + 2.05e5T^{2} \)
61 \( 1 - 107.T + 2.26e5T^{2} \)
67 \( 1 - 116.T + 3.00e5T^{2} \)
71 \( 1 + 139.T + 3.57e5T^{2} \)
73 \( 1 - 569.T + 3.89e5T^{2} \)
79 \( 1 + 309.T + 4.93e5T^{2} \)
83 \( 1 + 220.T + 5.71e5T^{2} \)
89 \( 1 + 1.37e3T + 7.04e5T^{2} \)
97 \( 1 + 1.10e3T + 9.12e5T^{2} \)
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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

−8.402247933471284642732697677457, −7.925422413107798875849688668016, −7.31802993529277220690410978057, −6.72638239069356451396570785732, −5.43767221205161096472194713263, −4.23201085786788462708797606137, −2.76331462229898833048984462323, −2.28113302753767874352075758283, −1.07435026645899194645769480794, 0, 1.07435026645899194645769480794, 2.28113302753767874352075758283, 2.76331462229898833048984462323, 4.23201085786788462708797606137, 5.43767221205161096472194713263, 6.72638239069356451396570785732, 7.31802993529277220690410978057, 7.925422413107798875849688668016, 8.402247933471284642732697677457

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