Properties

Label 2-1859-11.10-c0-0-7
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

\(L(\frac{1}{2})\) \(\approx\) \(2.111193160\)
\(L(\frac12)\) \(\approx\) \(2.111193160\)
\(L(1)\) 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

−9.408933157271934198136588809462, −8.501014710461866340974999531889, −7.85243625619789117458526318481, −7.36288778847386768586868736696, −6.19101814870035695047020715233, −5.65079930615928471228446956144, −4.29704197947666420624477345383, −3.22677663520357625486768732187, −2.48892331493762691856556133623, −1.82197688265522102554459922474, 1.82197688265522102554459922474, 2.48892331493762691856556133623, 3.22677663520357625486768732187, 4.29704197947666420624477345383, 5.65079930615928471228446956144, 6.19101814870035695047020715233, 7.36288778847386768586868736696, 7.85243625619789117458526318481, 8.501014710461866340974999531889, 9.408933157271934198136588809462

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