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Properties

Label 2-1860-1860.959-c0-0-3
Degree $2$
Conductor $1860$
Sign $0.569 + 0.822i$
Analytic cond. $0.928260$
Root an. cond. $0.963462$
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 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s + (−0.309 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)10-s + (0.309 + 0.951i)12-s + (−0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.690 − 0.951i)19-s + (0.309 − 0.951i)20-s + (−0.190 + 0.587i)23-s + (0.809 + 0.587i)24-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s + (−0.309 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)10-s + (0.309 + 0.951i)12-s + (−0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.690 − 0.951i)19-s + (0.309 − 0.951i)20-s + (−0.190 + 0.587i)23-s + (0.809 + 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.569 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643971246\)
\(L(\frac12)\) \(\approx\) \(1.643971246\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 - T \)
31 \( 1 + (0.809 + 0.587i)T \)
good7 \( 1 + (-0.809 + 0.587i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 - 1.17iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.382979756938008877586287180545, −9.171944892064632850199557686409, −7.41412188681288960369484987895, −6.55241758044160730898376861965, −5.82101165652754544591178294139, −5.26532189365900150104608408900, −4.49876845628826680620453637328, −3.50406459244607092231255741000, −2.46403116303247645554439243865, −1.18101840077494958566907930452, 1.62927469754151914494956834961, 2.65440979544589419323736901972, 3.92931043357240934054427857093, 5.07485738999547837359178308861, 5.54766106493140797177295970345, 6.27763249087299273818585944964, 6.92232958080548755146923910897, 7.69725640359054565877162172788, 8.540003043577932846906887954625, 9.539166228349723488487570672020

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