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Properties

Label 2-1800-24.11-c1-0-43
Degree $2$
Conductor $1800$
Sign $0.999 + 0.0413i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.06 + 0.927i)2-s + (0.280 − 1.98i)4-s + 3.02i·7-s + (1.53 + 2.37i)8-s − 3.62i·11-s + 1.69i·13-s + (−2.80 − 3.22i)14-s + (−3.84 − 1.11i)16-s − 6.60i·17-s − 5.12·19-s + (3.35 + 3.86i)22-s + 6.67·23-s + (−1.57 − 1.81i)26-s + (5.98 + 0.848i)28-s + 6.82·29-s + ⋯
L(s)  = 1  + (−0.755 + 0.655i)2-s + (0.140 − 0.990i)4-s + 1.14i·7-s + (0.543 + 0.839i)8-s − 1.09i·11-s + 0.470i·13-s + (−0.748 − 0.862i)14-s + (−0.960 − 0.277i)16-s − 1.60i·17-s − 1.17·19-s + (0.716 + 0.824i)22-s + 1.39·23-s + (−0.308 − 0.355i)26-s + (1.13 + 0.160i)28-s + 1.26·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0413i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.999 + 0.0413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.035521727\)
\(L(\frac12)\) \(\approx\) \(1.035521727\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 - 0.927i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 3.02iT - 7T^{2} \)
11 \( 1 + 3.62iT - 11T^{2} \)
13 \( 1 - 1.69iT - 13T^{2} \)
17 \( 1 + 6.60iT - 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 - 6.67T + 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + 0.371iT - 37T^{2} \)
41 \( 1 + 5.83iT - 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 + 0.525T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 4.86iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 2.45T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + 5.79iT - 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 - 9.33T + 97T^{2} \)
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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

−8.901083889814972748845905977182, −8.773012419473466943482303824075, −7.81454488709183629655619586228, −6.80546964880490332570792626709, −6.26563944904564437049134214838, −5.34128732662266560364217542703, −4.71097836816188127511967224481, −3.06670858985661247700901301379, −2.14679606383090411636532335955, −0.61524349033813813756643102878, 1.01208306966116548092836450514, 2.05514894492701819936459817677, 3.28090134856077847019662648645, 4.14924871961889076511330094223, 4.87480632782046826000194672874, 6.55892430084754899121475251081, 6.90135222231380385080939121244, 8.044342392926206806115233667241, 8.331106295593878033483056048435, 9.490461569428159261435057346905

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