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Properties

Label 2-1860-155.92-c1-0-26
Degree $2$
Conductor $1860$
Sign $-0.932 + 0.361i$
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 + (−0.198 + 2.22i)5-s + (0.548 + 0.548i)7-s + 1.00i·9-s − 2.53i·11-s + (−2.86 − 2.86i)13-s + (1.71 − 1.43i)15-s + (−2.54 + 2.54i)17-s + 2.84i·19-s − 0.775i·21-s + (−0.120 − 0.120i)23-s + (−4.92 − 0.885i)25-s + (0.707 − 0.707i)27-s + 0.666·29-s + (3.54 + 4.29i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.0888 + 0.996i)5-s + (0.207 + 0.207i)7-s + 0.333i·9-s − 0.763i·11-s + (−0.795 − 0.795i)13-s + (0.442 − 0.370i)15-s + (−0.617 + 0.617i)17-s + 0.652i·19-s − 0.169i·21-s + (−0.0252 − 0.0252i)23-s + (−0.984 − 0.177i)25-s + (0.136 − 0.136i)27-s + 0.123·29-s + (0.636 + 0.771i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.932 + 0.361i$
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: \((2,\ 1860,\ (\ :1/2),\ -0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1508139096\)
\(L(\frac12)\) \(\approx\) \(0.1508139096\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.198 - 2.22i)T \)
31 \( 1 + (-3.54 - 4.29i)T \)
good7 \( 1 + (-0.548 - 0.548i)T + 7iT^{2} \)
11 \( 1 + 2.53iT - 11T^{2} \)
13 \( 1 + (2.86 + 2.86i)T + 13iT^{2} \)
17 \( 1 + (2.54 - 2.54i)T - 17iT^{2} \)
19 \( 1 - 2.84iT - 19T^{2} \)
23 \( 1 + (0.120 + 0.120i)T + 23iT^{2} \)
29 \( 1 - 0.666T + 29T^{2} \)
37 \( 1 + (-1.99 + 1.99i)T - 37iT^{2} \)
41 \( 1 + 8.89T + 41T^{2} \)
43 \( 1 + (7.10 + 7.10i)T + 43iT^{2} \)
47 \( 1 + (3.40 + 3.40i)T + 47iT^{2} \)
53 \( 1 + (-2.68 - 2.68i)T + 53iT^{2} \)
59 \( 1 - 4.33iT - 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + (-0.114 - 0.114i)T + 67iT^{2} \)
71 \( 1 + 5.87T + 71T^{2} \)
73 \( 1 + (7.95 + 7.95i)T + 73iT^{2} \)
79 \( 1 + 1.76T + 79T^{2} \)
83 \( 1 + (-1.44 - 1.44i)T + 83iT^{2} \)
89 \( 1 + 6.41T + 89T^{2} \)
97 \( 1 + (11.1 + 11.1i)T + 97iT^{2} \)
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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

−8.632510176585912593581024969839, −8.110465089041528263355679981839, −7.20625663016572172156250245966, −6.54161307038425897991707886006, −5.78033562670782257551597505575, −4.98589681769602390681526311472, −3.71093670798415314892229943087, −2.84742747275000707830765778980, −1.78489281471255493318148244889, −0.05718005355863170568297924273, 1.44721957292730746450606470984, 2.68039397483950495327518021152, 4.22505404491953844107520071324, 4.62924743153382510014929672707, 5.27544150801955551490590462205, 6.45958689769322742375154320788, 7.17019328492055239065936053322, 8.080289914173491854215116935902, 8.940613994422760182877474596226, 9.630482997288849506025897829301

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