Properties

Label 2-1859-1.1-c3-0-258
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  − 0.631·2-s + 9.24·3-s − 7.60·4-s − 19.0·5-s − 5.83·6-s − 25.4·7-s + 9.84·8-s + 58.4·9-s + 12.0·10-s + 11·11-s − 70.2·12-s + 16.0·14-s − 176.·15-s + 54.5·16-s + 31.0·17-s − 36.8·18-s + 73.2·19-s + 144.·20-s − 235.·21-s − 6.94·22-s − 95.8·23-s + 91.0·24-s + 237.·25-s + 290.·27-s + 193.·28-s + 227.·29-s + 111.·30-s + ⋯
L(s)  = 1  − 0.223·2-s + 1.77·3-s − 0.950·4-s − 1.70·5-s − 0.396·6-s − 1.37·7-s + 0.435·8-s + 2.16·9-s + 0.380·10-s + 0.301·11-s − 1.69·12-s + 0.306·14-s − 3.03·15-s + 0.853·16-s + 0.443·17-s − 0.482·18-s + 0.884·19-s + 1.61·20-s − 2.44·21-s − 0.0672·22-s − 0.868·23-s + 0.774·24-s + 1.90·25-s + 2.07·27-s + 1.30·28-s + 1.45·29-s + 0.676·30-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: Trivial
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 \( 1 + 0.631T + 8T^{2} \)
3 \( 1 - 9.24T + 27T^{2} \)
5 \( 1 + 19.0T + 125T^{2} \)
7 \( 1 + 25.4T + 343T^{2} \)
17 \( 1 - 31.0T + 4.91e3T^{2} \)
19 \( 1 - 73.2T + 6.85e3T^{2} \)
23 \( 1 + 95.8T + 1.21e4T^{2} \)
29 \( 1 - 227.T + 2.43e4T^{2} \)
31 \( 1 + 5.28T + 2.97e4T^{2} \)
37 \( 1 - 304.T + 5.06e4T^{2} \)
41 \( 1 + 150.T + 6.89e4T^{2} \)
43 \( 1 + 172.T + 7.95e4T^{2} \)
47 \( 1 + 411.T + 1.03e5T^{2} \)
53 \( 1 - 185.T + 1.48e5T^{2} \)
59 \( 1 - 439.T + 2.05e5T^{2} \)
61 \( 1 + 507.T + 2.26e5T^{2} \)
67 \( 1 + 703.T + 3.00e5T^{2} \)
71 \( 1 + 519.T + 3.57e5T^{2} \)
73 \( 1 + 669.T + 3.89e5T^{2} \)
79 \( 1 + 42.5T + 4.93e5T^{2} \)
83 \( 1 - 57.1T + 5.71e5T^{2} \)
89 \( 1 - 521.T + 7.04e5T^{2} \)
97 \( 1 - 1.69e3T + 9.12e5T^{2} \)
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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

−8.555910986972992968175179394233, −7.80977532147937964326057979987, −7.43418542225003430167802210260, −6.37927748082168677448558189205, −4.74524574159804099576425877414, −3.99837613469223866951990917817, −3.41321585171673639942956161524, −2.90544296418134312531030111325, −1.13251301969491589052837298062, 0, 1.13251301969491589052837298062, 2.90544296418134312531030111325, 3.41321585171673639942956161524, 3.99837613469223866951990917817, 4.74524574159804099576425877414, 6.37927748082168677448558189205, 7.43418542225003430167802210260, 7.80977532147937964326057979987, 8.555910986972992968175179394233

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