Properties

Label 2-1815-1.1-c3-0-208
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  + 3.70·2-s + 3·3-s + 5.70·4-s + 5·5-s + 11.1·6-s − 9.10·7-s − 8.50·8-s + 9·9-s + 18.5·10-s + 17.1·12-s + 20.2·13-s − 33.7·14-s + 15·15-s − 77.1·16-s − 61.8·17-s + 33.3·18-s − 8.68·19-s + 28.5·20-s − 27.3·21-s − 127.·23-s − 25.5·24-s + 25·25-s + 75.1·26-s + 27·27-s − 51.9·28-s − 85.9·29-s + 55.5·30-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.577·3-s + 0.712·4-s + 0.447·5-s + 0.755·6-s − 0.491·7-s − 0.375·8-s + 0.333·9-s + 0.585·10-s + 0.411·12-s + 0.433·13-s − 0.643·14-s + 0.258·15-s − 1.20·16-s − 0.881·17-s + 0.436·18-s − 0.104·19-s + 0.318·20-s − 0.283·21-s − 1.15·23-s − 0.217·24-s + 0.200·25-s + 0.566·26-s + 0.192·27-s − 0.350·28-s − 0.550·29-s + 0.337·30-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 \( 1 - 3.70T + 8T^{2} \)
7 \( 1 + 9.10T + 343T^{2} \)
13 \( 1 - 20.2T + 2.19e3T^{2} \)
17 \( 1 + 61.8T + 4.91e3T^{2} \)
19 \( 1 + 8.68T + 6.85e3T^{2} \)
23 \( 1 + 127.T + 1.21e4T^{2} \)
29 \( 1 + 85.9T + 2.43e4T^{2} \)
31 \( 1 + 158.T + 2.97e4T^{2} \)
37 \( 1 - 138.T + 5.06e4T^{2} \)
41 \( 1 - 28.2T + 6.89e4T^{2} \)
43 \( 1 - 265.T + 7.95e4T^{2} \)
47 \( 1 + 515.T + 1.03e5T^{2} \)
53 \( 1 + 466.T + 1.48e5T^{2} \)
59 \( 1 - 373.T + 2.05e5T^{2} \)
61 \( 1 - 84.8T + 2.26e5T^{2} \)
67 \( 1 + 88.7T + 3.00e5T^{2} \)
71 \( 1 + 536.T + 3.57e5T^{2} \)
73 \( 1 - 322.T + 3.89e5T^{2} \)
79 \( 1 - 265.T + 4.93e5T^{2} \)
83 \( 1 + 520.T + 5.71e5T^{2} \)
89 \( 1 + 464.T + 7.04e5T^{2} \)
97 \( 1 + 204.T + 9.12e5T^{2} \)
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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

−8.598589875853202426457129744490, −7.60242878441684910899767958349, −6.53891211914455939643634306485, −6.09918410394423305947878629641, −5.16260555057876345865275581251, −4.24025502974766166786762098241, −3.58148254422038250591354585898, −2.68098720031604988372088964112, −1.78958960317689192300296613911, 0, 1.78958960317689192300296613911, 2.68098720031604988372088964112, 3.58148254422038250591354585898, 4.24025502974766166786762098241, 5.16260555057876345865275581251, 6.09918410394423305947878629641, 6.53891211914455939643634306485, 7.60242878441684910899767958349, 8.598589875853202426457129744490

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