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Properties

Label 2-1800-40.29-c1-0-70
Degree $2$
Conductor $1800$
Sign $0.831 + 0.555i$
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  + (1.41 + 0.0591i)2-s + (1.99 + 0.167i)4-s − 1.33i·7-s + (2.80 + 0.353i)8-s − 2.94i·11-s + 2.04·13-s + (0.0788 − 1.88i)14-s + (3.94 + 0.665i)16-s + 3.61i·17-s − 5.35i·19-s + (0.174 − 4.16i)22-s − 8.59i·23-s + (2.88 + 0.120i)26-s + (0.222 − 2.65i)28-s + 5.26i·29-s + ⋯
L(s)  = 1  + (0.999 + 0.0418i)2-s + (0.996 + 0.0835i)4-s − 0.504i·7-s + (0.992 + 0.125i)8-s − 0.887i·11-s + 0.566·13-s + (0.0210 − 0.503i)14-s + (0.986 + 0.166i)16-s + 0.876i·17-s − 1.22i·19-s + (0.0371 − 0.886i)22-s − 1.79i·23-s + (0.565 + 0.0236i)26-s + (0.0421 − 0.502i)28-s + 0.977i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.831 + 0.555i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.540078323\)
\(L(\frac12)\) \(\approx\) \(3.540078323\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0591i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.33iT - 7T^{2} \)
11 \( 1 + 2.94iT - 11T^{2} \)
13 \( 1 - 2.04T + 13T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 + 5.35iT - 19T^{2} \)
23 \( 1 + 8.59iT - 23T^{2} \)
29 \( 1 - 5.26iT - 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 - 6.55T + 37T^{2} \)
41 \( 1 + 7.02T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
47 \( 1 - 9.97iT - 47T^{2} \)
53 \( 1 + 6.12T + 53T^{2} \)
59 \( 1 - 4.75iT - 59T^{2} \)
61 \( 1 + 8.51iT - 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 2.62T + 71T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−9.059419210315472315774414096799, −8.321697922080244445179240804977, −7.47390150032846896145634356833, −6.52197440202637834054980258789, −6.09929837528087983635282021154, −5.00657903421909132691935605696, −4.24022928424128338568875889150, −3.38531991105186647898205878567, −2.45947582666345287309502679523, −1.02534780480274398230549307153, 1.52478934687091071161876703487, 2.49414745681258587922575542311, 3.58688213403963566437687937483, 4.32588212885652445966652418968, 5.44106630957428218721531070130, 5.82063187127772620993937860877, 6.93111880535797812675819091700, 7.54981575579481692829354808834, 8.406633555669126613942281906262, 9.629797653205517325879944397289

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