Properties

Label 2-1859-143.10-c0-0-0
Degree $2$
Conductor $1859$
Sign $0.964 - 0.265i$
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  + (−0.809 − 1.40i)2-s + (0.809 + 1.40i)3-s + (−0.809 + 1.40i)4-s + (1.30 − 2.26i)6-s + (0.309 − 0.535i)7-s + 0.999·8-s + (−0.809 + 1.40i)9-s + (0.5 + 0.866i)11-s − 2.61·12-s − 14-s + 2.61·18-s + (−0.809 + 1.40i)19-s + 21-s + (0.809 − 1.40i)22-s + (−0.309 − 0.535i)23-s + (0.809 + 1.40i)24-s + ⋯
L(s)  = 1  + (−0.809 − 1.40i)2-s + (0.809 + 1.40i)3-s + (−0.809 + 1.40i)4-s + (1.30 − 2.26i)6-s + (0.309 − 0.535i)7-s + 0.999·8-s + (−0.809 + 1.40i)9-s + (0.5 + 0.866i)11-s − 2.61·12-s − 14-s + 2.61·18-s + (−0.809 + 1.40i)19-s + 21-s + (0.809 − 1.40i)22-s + (−0.309 − 0.535i)23-s + (0.809 + 1.40i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.964 - 0.265i$
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: \((2,\ 1859,\ (\ :0),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9420899148\)
\(L(\frac12)\) \(\approx\) \(0.9420899148\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.618T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.751437454139908517872216840112, −8.933895856850619119019741596594, −8.372314642148071192982909755015, −7.63603151567236868656639011908, −6.29443076873478273602733820603, −4.78816060941540580756413394176, −4.16613542139994979735005208055, −3.52785419925560541736864567178, −2.56247211382127301731088944395, −1.56014463873826563548800012018, 0.897805846323087410129088445687, 2.19796428821317423964597392643, 3.26144336822844253568526095690, 4.87338470015644872633230847770, 5.90687373248853390396662274767, 6.53887047486827115583102688257, 7.10476122860471875531427344265, 7.84025876069348188935555164022, 8.581750714540021898692226562912, 8.814501259102035688436781298781

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