Properties

Label 2-1859-1.1-c1-0-85
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $14.8441$
Root an. cond. $3.85281$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.60·3-s − 4-s − 3.76·5-s − 2.60·6-s − 0.167·7-s + 3·8-s + 3.76·9-s + 3.76·10-s − 11-s − 2.60·12-s + 0.167·14-s − 9.80·15-s − 16-s + 4.37·17-s − 3.76·18-s + 3.37·19-s + 3.76·20-s − 0.434·21-s + 22-s − 5.20·23-s + 7.80·24-s + 9.20·25-s + 2.00·27-s + 0.167·28-s + 1.93·29-s + 9.80·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.50·3-s − 0.5·4-s − 1.68·5-s − 1.06·6-s − 0.0631·7-s + 1.06·8-s + 1.25·9-s + 1.19·10-s − 0.301·11-s − 0.751·12-s + 0.0446·14-s − 2.53·15-s − 0.250·16-s + 1.05·17-s − 0.888·18-s + 0.773·19-s + 0.842·20-s − 0.0948·21-s + 0.213·22-s − 1.08·23-s + 1.59·24-s + 1.84·25-s + 0.384·27-s + 0.0315·28-s + 0.359·29-s + 1.79·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(14.8441\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 + 3.76T + 5T^{2} \)
7 \( 1 + 0.167T + 7T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 + 5.20T + 23T^{2} \)
29 \( 1 - 1.93T + 29T^{2} \)
31 \( 1 + 6.86T + 31T^{2} \)
37 \( 1 + 8.63T + 37T^{2} \)
41 \( 1 - 1.26T + 41T^{2} \)
43 \( 1 + 7.20T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 1.33T + 53T^{2} \)
59 \( 1 + 2.13T + 59T^{2} \)
61 \( 1 + 4.80T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 3.56T + 97T^{2} \)
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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

−8.538298757838176634606157430030, −8.300750352023258414043192083396, −7.48030421733173809924907587768, −7.27826251990656497401388546419, −5.43716698325643619821817027044, −4.34202510139997404983480576011, −3.67854344988793548825357380144, −3.04576354798957517936371072780, −1.52211600501641223530039251974, 0, 1.52211600501641223530039251974, 3.04576354798957517936371072780, 3.67854344988793548825357380144, 4.34202510139997404983480576011, 5.43716698325643619821817027044, 7.27826251990656497401388546419, 7.48030421733173809924907587768, 8.300750352023258414043192083396, 8.538298757838176634606157430030

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