Properties

Label 2-1859-11.10-c0-0-6
Degree $2$
Conductor $1859$
Sign $1$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.24·3-s + 4-s − 0.445·5-s + 0.554·9-s + 11-s + 1.24·12-s − 0.554·15-s + 16-s − 0.445·20-s − 1.80·23-s − 0.801·25-s − 0.554·27-s − 1.80·31-s + 1.24·33-s + 0.554·36-s + 1.24·37-s + 44-s − 0.246·45-s − 1.80·47-s + 1.24·48-s + 49-s − 0.445·53-s − 0.445·55-s + 1.24·59-s − 0.554·60-s + 64-s + 2·67-s + ⋯
L(s)  = 1  + 1.24·3-s + 4-s − 0.445·5-s + 0.554·9-s + 11-s + 1.24·12-s − 0.554·15-s + 16-s − 0.445·20-s − 1.80·23-s − 0.801·25-s − 0.554·27-s − 1.80·31-s + 1.24·33-s + 0.554·36-s + 1.24·37-s + 44-s − 0.246·45-s − 1.80·47-s + 1.24·48-s + 49-s − 0.445·53-s − 0.445·55-s + 1.24·59-s − 0.554·60-s + 64-s + 2·67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (846, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.990501239\)
\(L(\frac12)\) \(\approx\) \(1.990501239\)
\(L(1)\) 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 \( 1 - T^{2} \)
3 \( 1 - 1.24T + T^{2} \)
5 \( 1 + 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.80T + T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 - 1.24T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 - 1.24T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−9.501240855799895144283972477294, −8.415793773469297177210293674414, −7.961988389777494940333401401128, −7.24302629205076807798147706619, −6.40568753789253779618788994161, −5.56659645466655133035878205957, −3.95510519673818623311143131366, −3.63519578290581189004729671776, −2.45652022995288226292771721013, −1.71579201571265490817666154187, 1.71579201571265490817666154187, 2.45652022995288226292771721013, 3.63519578290581189004729671776, 3.95510519673818623311143131366, 5.56659645466655133035878205957, 6.40568753789253779618788994161, 7.24302629205076807798147706619, 7.961988389777494940333401401128, 8.415793773469297177210293674414, 9.501240855799895144283972477294

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