Properties

Label 2-1859-1.1-c3-0-110
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $109.684$
Root an. cond. $10.4730$
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  − 2.73·2-s − 7.92·3-s − 0.535·4-s − 14.8·5-s + 21.6·6-s − 3.07·7-s + 23.3·8-s + 35.8·9-s + 40.5·10-s + 11·11-s + 4.24·12-s + 8.39·14-s + 117.·15-s − 59.4·16-s − 41.2·17-s − 97.9·18-s − 139.·19-s + 7.96·20-s + 24.3·21-s − 30.0·22-s − 111.·23-s − 184.·24-s + 95.7·25-s − 70.2·27-s + 1.64·28-s − 24.9·29-s − 321.·30-s + ⋯
L(s)  = 1  − 0.965·2-s − 1.52·3-s − 0.0669·4-s − 1.32·5-s + 1.47·6-s − 0.165·7-s + 1.03·8-s + 1.32·9-s + 1.28·10-s + 0.301·11-s + 0.102·12-s + 0.160·14-s + 2.02·15-s − 0.928·16-s − 0.588·17-s − 1.28·18-s − 1.68·19-s + 0.0890·20-s + 0.253·21-s − 0.291·22-s − 1.00·23-s − 1.57·24-s + 0.765·25-s − 0.500·27-s + 0.0111·28-s − 0.160·29-s − 1.95·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(109.684\)
Root analytic conductor: \(10.4730\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :3/2),\ -1)\)

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)$
bad11 \( 1 - 11T \)
13 \( 1 \)
good2 \( 1 + 2.73T + 8T^{2} \)
3 \( 1 + 7.92T + 27T^{2} \)
5 \( 1 + 14.8T + 125T^{2} \)
7 \( 1 + 3.07T + 343T^{2} \)
17 \( 1 + 41.2T + 4.91e3T^{2} \)
19 \( 1 + 139.T + 6.85e3T^{2} \)
23 \( 1 + 111.T + 1.21e4T^{2} \)
29 \( 1 + 24.9T + 2.43e4T^{2} \)
31 \( 1 + 31.4T + 2.97e4T^{2} \)
37 \( 1 + 13.1T + 5.06e4T^{2} \)
41 \( 1 + 261.T + 6.89e4T^{2} \)
43 \( 1 + 57.7T + 7.95e4T^{2} \)
47 \( 1 - 343.T + 1.03e5T^{2} \)
53 \( 1 + 342.T + 1.48e5T^{2} \)
59 \( 1 + 88.3T + 2.05e5T^{2} \)
61 \( 1 - 738.T + 2.26e5T^{2} \)
67 \( 1 + 342.T + 3.00e5T^{2} \)
71 \( 1 - 207.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 1.29e3T + 4.93e5T^{2} \)
83 \( 1 + 441.T + 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
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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

−8.384311297943294779697285171068, −7.86254375154807664653974647401, −6.85517183903682839525648121883, −6.37235020052196988075778988231, −5.17393220382415998759618632836, −4.37484778090141111958558376645, −3.81641574227619528642928317415, −1.90703160572882329407934455871, −0.59039637703878272589947666685, 0, 0.59039637703878272589947666685, 1.90703160572882329407934455871, 3.81641574227619528642928317415, 4.37484778090141111958558376645, 5.17393220382415998759618632836, 6.37235020052196988075778988231, 6.85517183903682839525648121883, 7.86254375154807664653974647401, 8.384311297943294779697285171068

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