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Properties

Label 2-1860-155.123-c1-0-19
Degree $2$
Conductor $1860$
Sign $0.353 + 0.935i$
Analytic cond. $14.8521$
Root an. cond. $3.85385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.193 − 2.22i)5-s + (−2.45 + 2.45i)7-s − 1.00i·9-s + 2.71i·11-s + (3.28 − 3.28i)13-s + (1.71 + 1.43i)15-s + (−1.47 − 1.47i)17-s + 2.29i·19-s − 3.47i·21-s + (1.10 − 1.10i)23-s + (−4.92 + 0.863i)25-s + (0.707 + 0.707i)27-s + 1.82·29-s + (−5.56 + 0.237i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.0867 − 0.996i)5-s + (−0.928 + 0.928i)7-s − 0.333i·9-s + 0.818i·11-s + (0.910 − 0.910i)13-s + (0.442 + 0.371i)15-s + (−0.357 − 0.357i)17-s + 0.525i·19-s − 0.758i·21-s + (0.230 − 0.230i)23-s + (−0.984 + 0.172i)25-s + (0.136 + 0.136i)27-s + 0.338·29-s + (−0.999 + 0.0427i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9435216966\)
\(L(\frac12)\) \(\approx\) \(0.9435216966\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.193 + 2.22i)T \)
31 \( 1 + (5.56 - 0.237i)T \)
good7 \( 1 + (2.45 - 2.45i)T - 7iT^{2} \)
11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 + (-3.28 + 3.28i)T - 13iT^{2} \)
17 \( 1 + (1.47 + 1.47i)T + 17iT^{2} \)
19 \( 1 - 2.29iT - 19T^{2} \)
23 \( 1 + (-1.10 + 1.10i)T - 23iT^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
37 \( 1 + (4.28 + 4.28i)T + 37iT^{2} \)
41 \( 1 - 6.87T + 41T^{2} \)
43 \( 1 + (-4.47 + 4.47i)T - 43iT^{2} \)
47 \( 1 + (1.11 - 1.11i)T - 47iT^{2} \)
53 \( 1 + (-6.61 + 6.61i)T - 53iT^{2} \)
59 \( 1 + 6.91iT - 59T^{2} \)
61 \( 1 + 9.59iT - 61T^{2} \)
67 \( 1 + (-0.811 + 0.811i)T - 67iT^{2} \)
71 \( 1 + 0.0283T + 71T^{2} \)
73 \( 1 + (-6.32 + 6.32i)T - 73iT^{2} \)
79 \( 1 + 8.59T + 79T^{2} \)
83 \( 1 + (-10.5 + 10.5i)T - 83iT^{2} \)
89 \( 1 - 1.53T + 89T^{2} \)
97 \( 1 + (-6.71 + 6.71i)T - 97iT^{2} \)
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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

−9.137543006489725211227398015902, −8.540763710727543329588804625006, −7.57598965323281254335053385112, −6.50836192380817755009154835963, −5.70365675845044534200420955458, −5.19067314264756248795128452768, −4.14929691053323839023648453684, −3.27988265159466892542844033539, −1.98242533230005354909959807550, −0.42858122143032527807365647976, 1.07252817612339581426939940274, 2.55943728351964112253368651634, 3.56764679645797516512857230044, 4.20221617206472608093433294415, 5.68344881429776858036020813430, 6.39457309639479791988504829730, 6.88690595801731051650599184061, 7.54700817734476967884959138094, 8.618968762341671012929143026642, 9.412747593842462746456726502939

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