Properties

Label 2-1859-1.1-c1-0-117
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $14.8441$
Root an. cond. $3.85281$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 2·7-s − 2·9-s − 2·10-s − 11-s − 2·12-s + 4·14-s + 15-s − 4·16-s − 2·17-s − 4·18-s − 2·20-s − 2·21-s − 2·22-s − 23-s − 4·25-s + 5·27-s + 4·28-s + 2·30-s − 7·31-s − 8·32-s + 33-s − 4·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s − 0.632·10-s − 0.301·11-s − 0.577·12-s + 1.06·14-s + 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s − 0.447·20-s − 0.436·21-s − 0.426·22-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 0.755·28-s + 0.365·30-s − 1.25·31-s − 1.41·32-s + 0.174·33-s − 0.685·34-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: yes
Arithmetic: yes
Character: Trivial
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 \( 1 - p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.679083346749551349707825193313, −7.972094664118160382294418175959, −6.98822017919486224475963260290, −6.13551737647869362536665067533, −5.42467290200223196482028296291, −4.84072493234829119227722123670, −4.01156608984212096591011250141, −3.11506443817379691538930485019, −1.98576135654663457689533375342, 0, 1.98576135654663457689533375342, 3.11506443817379691538930485019, 4.01156608984212096591011250141, 4.84072493234829119227722123670, 5.42467290200223196482028296291, 6.13551737647869362536665067533, 6.98822017919486224475963260290, 7.972094664118160382294418175959, 8.679083346749551349707825193313

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