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Properties

Label 2-1859-143.10-c0-0-6
Degree $2$
Conductor $1859$
Sign $-0.500 + 0.865i$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 1.07i)3-s + (0.5 − 0.866i)4-s + 0.445i·5-s + (−0.277 + 0.480i)9-s + (0.866 − 0.5i)11-s − 1.24·12-s + (0.480 − 0.277i)15-s + (−0.499 − 0.866i)16-s + (0.385 + 0.222i)20-s + (−0.900 − 1.56i)23-s + 0.801·25-s − 0.554·27-s + 1.80i·31-s + (−1.07 − 0.623i)33-s + (0.277 + 0.480i)36-s + (1.07 − 0.623i)37-s + ⋯
L(s)  = 1  + (−0.623 − 1.07i)3-s + (0.5 − 0.866i)4-s + 0.445i·5-s + (−0.277 + 0.480i)9-s + (0.866 − 0.5i)11-s − 1.24·12-s + (0.480 − 0.277i)15-s + (−0.499 − 0.866i)16-s + (0.385 + 0.222i)20-s + (−0.900 − 1.56i)23-s + 0.801·25-s − 0.554·27-s + 1.80i·31-s + (−1.07 − 0.623i)33-s + (0.277 + 0.480i)36-s + (1.07 − 0.623i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.500 + 0.865i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.500 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.039886811\)
\(L(\frac12)\) \(\approx\) \(1.039886811\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 0.445iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.80iT - T^{2} \)
37 \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + 1.80iT - T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.165248408307395126073709361120, −8.307845132598080543566015954522, −7.19496784341395253020123913096, −6.69591412919094952735167305185, −6.20759881250172895375592883475, −5.47282035065815296907287183926, −4.32106616501331345575328965618, −2.94883471590125142291448431570, −1.86476893458269782276525102609, −0.870017456613890120187473699139, 1.74733066374949716477037617407, 3.12559424009857870559348240436, 4.14882783878984780518357670631, 4.50212477317600144408087014487, 5.68577477017447683745931105928, 6.37082529711421098734855626882, 7.47422898765357525773462722295, 8.016381530844167035622334690399, 9.222835610204553926362171361570, 9.527016631462661434793469763254

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