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Properties

Label 2-1860-155.92-c1-0-5
Degree $2$
Conductor $1860$
Sign $-0.0302 - 0.999i$
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 + (0.773 − 2.09i)5-s + (−0.100 − 0.100i)7-s + 1.00i·9-s + 6.12i·11-s + (1.74 + 1.74i)13-s + (2.03 − 0.936i)15-s + (−4.02 + 4.02i)17-s + 4.45i·19-s − 0.141i·21-s + (−2.71 − 2.71i)23-s + (−3.80 − 3.24i)25-s + (−0.707 + 0.707i)27-s − 0.134·29-s + (−4.54 + 3.21i)31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.345 − 0.938i)5-s + (−0.0378 − 0.0378i)7-s + 0.333i·9-s + 1.84i·11-s + (0.484 + 0.484i)13-s + (0.524 − 0.241i)15-s + (−0.976 + 0.976i)17-s + 1.02i·19-s − 0.0309i·21-s + (−0.566 − 0.566i)23-s + (−0.760 − 0.648i)25-s + (−0.136 + 0.136i)27-s − 0.0248·29-s + (−0.816 + 0.577i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0302 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0302 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0302 - 0.999i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ -0.0302 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.704490605\)
\(L(\frac12)\) \(\approx\) \(1.704490605\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.773 + 2.09i)T \)
31 \( 1 + (4.54 - 3.21i)T \)
good7 \( 1 + (0.100 + 0.100i)T + 7iT^{2} \)
11 \( 1 - 6.12iT - 11T^{2} \)
13 \( 1 + (-1.74 - 1.74i)T + 13iT^{2} \)
17 \( 1 + (4.02 - 4.02i)T - 17iT^{2} \)
19 \( 1 - 4.45iT - 19T^{2} \)
23 \( 1 + (2.71 + 2.71i)T + 23iT^{2} \)
29 \( 1 + 0.134T + 29T^{2} \)
37 \( 1 + (-3.24 + 3.24i)T - 37iT^{2} \)
41 \( 1 - 0.335T + 41T^{2} \)
43 \( 1 + (0.907 + 0.907i)T + 43iT^{2} \)
47 \( 1 + (-0.662 - 0.662i)T + 47iT^{2} \)
53 \( 1 + (-3.93 - 3.93i)T + 53iT^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + (-5.93 - 5.93i)T + 67iT^{2} \)
71 \( 1 - 1.48T + 71T^{2} \)
73 \( 1 + (8.06 + 8.06i)T + 73iT^{2} \)
79 \( 1 - 6.67T + 79T^{2} \)
83 \( 1 + (-9.79 - 9.79i)T + 83iT^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + (-0.653 - 0.653i)T + 97iT^{2} \)
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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

−9.313906183288418791678795925168, −8.804492666751351831446838148340, −8.035328740164418427161624960043, −7.14159066261295728128978567340, −6.19717779772879315723539898000, −5.29022576254368024968033083212, −4.26196117065988063805804910977, −4.02412260119390580189481164062, −2.26004434229178244301195610019, −1.61733904103632802882849249439, 0.57016529620320237501514651265, 2.16656473236145942019662245651, 3.04636267368294580402842051014, 3.66729973043407799887967355787, 5.12220717410756485858697420491, 6.08979445272454462657604978633, 6.54901056131520229775017976249, 7.50218048789373505878481772044, 8.211977906980086965865962339392, 9.058193958261695211314741041408

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