↑

Properties

Label 2-1860-155.92-c1-0-1
Degree $2$
Conductor $1860$
Sign $0.475 - 0.879i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.475 - 0.879i$
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: \((2,\ 1860,\ (\ :1/2),\ 0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5625838389\)
\(L(\frac12)\) \(\approx\) \(0.5625838389\)
\(L(\frac{3}{2})\) 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 \( 1 + (-11.3 - 11.3i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.566821410258502404356248827907, −8.357102873272240875232782599295, −7.87773759628772339345266030162, −6.97426915649178890144275515051, −6.28162190441046733156032748189, −5.15580948448178641516678253234, −4.68938483019377794525571427297, −3.59586780814952735047460541579, −2.26494654134724653035424921005, −1.06668371244780648173651039192, 0.24558112409870220794965911267, 2.48734025987923151419537306901, 2.91571127170337900105652752155, 4.21255963194128084819246511126, 4.98760870959546621693325624695, 6.05656198763752667172263628781, 6.72065302584764911736801701286, 7.19354072132889813889111189591, 8.518935172886629877484173918315, 9.271733241175432519355868494597

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