Properties

Label 2-1800-40.29-c1-0-6
Degree $2$
Conductor $1800$
Sign $-0.988 + 0.151i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  + (0.576 + 1.29i)2-s + (−1.33 + 1.48i)4-s − 1.97i·7-s + (−2.69 − 0.867i)8-s − 1.43i·11-s − 0.241·13-s + (2.55 − 1.13i)14-s + (−0.430 − 3.97i)16-s + 7.38i·17-s + 3.04i·19-s + (1.84 − 0.824i)22-s + 0.874i·23-s + (−0.139 − 0.311i)26-s + (2.94 + 2.64i)28-s + 9.07i·29-s + ⋯
L(s)  = 1  + (0.407 + 0.913i)2-s + (−0.667 + 0.744i)4-s − 0.747i·7-s + (−0.951 − 0.306i)8-s − 0.431i·11-s − 0.0669·13-s + (0.682 − 0.304i)14-s + (−0.107 − 0.994i)16-s + 1.79i·17-s + 0.697i·19-s + (0.393 − 0.175i)22-s + 0.182i·23-s + (−0.0272 − 0.0611i)26-s + (0.556 + 0.499i)28-s + 1.68i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.988 + 0.151i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9624951412\)
\(L(\frac12)\) \(\approx\) \(0.9624951412\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.576 - 1.29i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 + 0.241T + 13T^{2} \)
17 \( 1 - 7.38iT - 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 - 0.874iT - 23T^{2} \)
29 \( 1 - 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 + 8.81T + 37T^{2} \)
41 \( 1 - 1.91T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 3.34iT - 47T^{2} \)
53 \( 1 + 9.20T + 53T^{2} \)
59 \( 1 - 6.43iT - 59T^{2} \)
61 \( 1 + 4.57iT - 61T^{2} \)
67 \( 1 + 4.86T + 67T^{2} \)
71 \( 1 - 8.21T + 71T^{2} \)
73 \( 1 - 4.12iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 8.08T + 89T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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

−9.501108462954779310559946218411, −8.616626819648030613939321380139, −8.074654429630597436677951684189, −7.21913669179024954247699292499, −6.53484005170117505147780743426, −5.71174087255496602064505950417, −4.93408782516030075478024044683, −3.79492375559801229536990878788, −3.42090038705494174920182773690, −1.58406532813110328048032317199, 0.30306028359456781909766979061, 1.93720927116093820843822518833, 2.70218802917496715084675988820, 3.67106729698652827787523061739, 4.85737933937018859555440413004, 5.23825129548562015207396339565, 6.31634886949073098686915874285, 7.21257893910511068393589617650, 8.293912767519137788204069331659, 9.251965472587226607043877949772

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