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Properties

Label 2-1860-155.123-c1-0-10
Degree $2$
Conductor $1860$
Sign $0.159 - 0.987i$
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 + (−0.143 + 2.23i)5-s + (−1.60 + 1.60i)7-s − 1.00i·9-s + 3.63i·11-s + (4.58 − 4.58i)13-s + (1.47 + 1.67i)15-s + (3.60 + 3.60i)17-s − 4.61i·19-s + 2.27i·21-s + (−1.92 + 1.92i)23-s + (−4.95 − 0.640i)25-s + (−0.707 − 0.707i)27-s − 6.98·29-s + (5.27 + 1.79i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.0641 + 0.997i)5-s + (−0.606 + 0.606i)7-s − 0.333i·9-s + 1.09i·11-s + (1.27 − 1.27i)13-s + (0.381 + 0.433i)15-s + (0.874 + 0.874i)17-s − 1.05i·19-s + 0.495i·21-s + (−0.402 + 0.402i)23-s + (−0.991 − 0.128i)25-s + (−0.136 − 0.136i)27-s − 1.29·29-s + (0.946 + 0.321i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.159 - 0.987i$
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: \((2,\ 1860,\ (\ :1/2),\ 0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709119904\)
\(L(\frac12)\) \(\approx\) \(1.709119904\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.143 - 2.23i)T \)
31 \( 1 + (-5.27 - 1.79i)T \)
good7 \( 1 + (1.60 - 1.60i)T - 7iT^{2} \)
11 \( 1 - 3.63iT - 11T^{2} \)
13 \( 1 + (-4.58 + 4.58i)T - 13iT^{2} \)
17 \( 1 + (-3.60 - 3.60i)T + 17iT^{2} \)
19 \( 1 + 4.61iT - 19T^{2} \)
23 \( 1 + (1.92 - 1.92i)T - 23iT^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
37 \( 1 + (-5.97 - 5.97i)T + 37iT^{2} \)
41 \( 1 - 2.78T + 41T^{2} \)
43 \( 1 + (3.64 - 3.64i)T - 43iT^{2} \)
47 \( 1 + (6.61 - 6.61i)T - 47iT^{2} \)
53 \( 1 + (2.45 - 2.45i)T - 53iT^{2} \)
59 \( 1 - 12.7iT - 59T^{2} \)
61 \( 1 - 9.01iT - 61T^{2} \)
67 \( 1 + (6.84 - 6.84i)T - 67iT^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (3.05 - 3.05i)T - 73iT^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + (-0.881 + 0.881i)T - 83iT^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (-11.3 + 11.3i)T - 97iT^{2} \)
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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

−9.538493382026022920754020452921, −8.488989618302391053480394855652, −7.81198503472000670709244320835, −7.12389806064713466427840159778, −6.15879789146175637308349547408, −5.78132188170329169765392060394, −4.29920636524296283772254254831, −3.21763391582290517183272599572, −2.74957324567372714057351562210, −1.42489018938060595312677708164, 0.62579903782143796219448226458, 1.88565670106663549629040636414, 3.60026827585449684057723376048, 3.74448148764365304010928922050, 4.90595815364003567217426419919, 5.86427975407227529322274667157, 6.54827507896866476880720743680, 7.81045579870610168778207074959, 8.276705585628034782111689350751, 9.201539262321900975742470541819

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