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Properties

Label 2-1860-155.123-c1-0-13
Degree $2$
Conductor $1860$
Sign $0.993 + 0.110i$
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 + (−1.72 − 1.41i)5-s + (−0.575 + 0.575i)7-s − 1.00i·9-s + 5.07i·11-s + (−0.293 + 0.293i)13-s + (−2.22 + 0.218i)15-s + (0.365 + 0.365i)17-s + 0.439i·19-s + 0.814i·21-s + (4.26 − 4.26i)23-s + (0.972 + 4.90i)25-s + (−0.707 − 0.707i)27-s + 6.60·29-s + (5.23 − 1.88i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.772 − 0.634i)5-s + (−0.217 + 0.217i)7-s − 0.333i·9-s + 1.53i·11-s + (−0.0813 + 0.0813i)13-s + (−0.574 + 0.0563i)15-s + (0.0887 + 0.0887i)17-s + 0.100i·19-s + 0.177i·21-s + (0.889 − 0.889i)23-s + (0.194 + 0.980i)25-s + (−0.136 − 0.136i)27-s + 1.22·29-s + (0.940 − 0.338i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.993 + 0.110i$
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: \((2,\ 1860,\ (\ :1/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649727627\)
\(L(\frac12)\) \(\approx\) \(1.649727627\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 1.41i)T \)
31 \( 1 + (-5.23 + 1.88i)T \)
good7 \( 1 + (0.575 - 0.575i)T - 7iT^{2} \)
11 \( 1 - 5.07iT - 11T^{2} \)
13 \( 1 + (0.293 - 0.293i)T - 13iT^{2} \)
17 \( 1 + (-0.365 - 0.365i)T + 17iT^{2} \)
19 \( 1 - 0.439iT - 19T^{2} \)
23 \( 1 + (-4.26 + 4.26i)T - 23iT^{2} \)
29 \( 1 - 6.60T + 29T^{2} \)
37 \( 1 + (-0.849 - 0.849i)T + 37iT^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + (6.93 - 6.93i)T - 43iT^{2} \)
47 \( 1 + (3.89 - 3.89i)T - 47iT^{2} \)
53 \( 1 + (-8.89 + 8.89i)T - 53iT^{2} \)
59 \( 1 + 8.14iT - 59T^{2} \)
61 \( 1 + 1.48iT - 61T^{2} \)
67 \( 1 + (6.46 - 6.46i)T - 67iT^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + (-10.0 + 10.0i)T - 73iT^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-4.35 + 4.35i)T - 83iT^{2} \)
89 \( 1 - 4.17T + 89T^{2} \)
97 \( 1 + (5.28 - 5.28i)T - 97iT^{2} \)
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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

−9.188065596923223905919150194672, −8.326874603536662363301873109816, −7.76127769273814209956097557078, −6.96006845726269082851413470053, −6.23187219400690710119318478698, −4.81057206743589762679779647800, −4.48355981456787564206637802734, −3.23751900659297568245885975505, −2.23782681974924392314294606122, −0.954670880217869497754514819444, 0.78409481668804861795968206459, 2.70330258654028786801418787393, 3.30597649529512423085010736951, 4.07320482072243226775328367873, 5.12941563046836933847115973849, 6.12507572861728702335225605648, 6.97108006508000324614275503753, 7.75414587689629646096134996422, 8.513271112995941492671903773371, 9.074477457836922925747606249052

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