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Properties

Label 2-1859-143.43-c0-0-4
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  + (0.309 − 0.535i)2-s + (−0.309 + 0.535i)3-s + (0.309 + 0.535i)4-s + (0.190 + 0.330i)6-s + (−0.809 − 1.40i)7-s + 0.999·8-s + (0.309 + 0.535i)9-s + (0.5 − 0.866i)11-s − 0.381·12-s − 14-s + 0.381·18-s + (0.309 + 0.535i)19-s + 21-s + (−0.309 − 0.535i)22-s + (0.809 − 1.40i)23-s + (−0.309 + 0.535i)24-s + ⋯
L(s)  = 1  + (0.309 − 0.535i)2-s + (−0.309 + 0.535i)3-s + (0.309 + 0.535i)4-s + (0.190 + 0.330i)6-s + (−0.809 − 1.40i)7-s + 0.999·8-s + (0.309 + 0.535i)9-s + (0.5 − 0.866i)11-s − 0.381·12-s − 14-s + 0.381·18-s + (0.309 + 0.535i)19-s + 21-s + (−0.309 − 0.535i)22-s + (0.809 − 1.40i)23-s + (−0.309 + 0.535i)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: $\chi_{1859} (1330, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 0.964 + 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381345118\)
\(L(\frac12)\) \(\approx\) \(1.381345118\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
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 \( 1 + 0.618T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.61T + T^{2} \)
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

−9.688242645501371632220770095548, −8.600917816504514829540414896601, −7.76787933143495944653112452260, −6.94722450545858766786650154676, −6.38495362338698665169902262518, −5.01803144777260685952954823991, −4.25849949411059740883304861699, −3.58571279108148243151393541180, −2.79834940735570818638491562373, −1.20626941916860746336992408420, 1.35514688192140236055118122844, 2.42655582557872478330818560558, 3.63255972702343510523761362683, 4.96060050314026756305114350412, 5.56225393262291710788002689167, 6.36562870835256555025457009635, 6.91789388936982228852827860508, 7.44892659256378597463855215934, 8.858960169227901867166573903788, 9.381981748916703566313029652126

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