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Properties

Label 2-1800-8.5-c1-0-12
Degree $2$
Conductor $1800$
Sign $-0.968 - 0.247i$
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  + (−0.806 + 1.16i)2-s + (−0.699 − 1.87i)4-s + 0.746·7-s + (2.74 + 0.699i)8-s + 5.36i·11-s − 2.92i·13-s + (−0.601 + 0.866i)14-s + (−3.02 + 2.62i)16-s − 2.13·17-s − 1.73i·19-s + (−6.22 − 4.32i)22-s − 7.49·23-s + (3.39 + 2.35i)26-s + (−0.521 − 1.39i)28-s + 6.74i·29-s + ⋯
L(s)  = 1  + (−0.570 + 0.821i)2-s + (−0.349 − 0.936i)4-s + 0.282·7-s + (0.968 + 0.247i)8-s + 1.61i·11-s − 0.811i·13-s + (−0.160 + 0.231i)14-s + (−0.755 + 0.655i)16-s − 0.517·17-s − 0.397i·19-s + (−1.32 − 0.921i)22-s − 1.56·23-s + (0.666 + 0.462i)26-s + (−0.0985 − 0.264i)28-s + 1.25i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.968 - 0.247i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6846605149\)
\(L(\frac12)\) \(\approx\) \(0.6846605149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.806 - 1.16i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 0.746T + 7T^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 + 2.92iT - 13T^{2} \)
17 \( 1 + 2.13T + 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 7.49T + 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 - 1.07iT - 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 + 7.72iT - 53T^{2} \)
59 \( 1 - 6.85iT - 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 - 7.44iT - 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 0.690T + 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 - 5.85iT - 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−9.635324413332405920110069051275, −8.752727095877017634280124558904, −7.956534545486888424338768991294, −7.37019800418911238813745682770, −6.60674810383686846460189140944, −5.73092808799545101988349928454, −4.82419883028832056854201420005, −4.19223671677264163353280112870, −2.51143122665767480818789581501, −1.39711787656363246138400327695, 0.31655856550930498529516483951, 1.70659724983369181578375303341, 2.70425727409420225912067512817, 3.80147044257754391156384624286, 4.42364223440674861755524224572, 5.76030227930369107789094157181, 6.51270874828501657155697287671, 7.75250847267766898672574394383, 8.207565710786043607985568473507, 8.995210991416337013539134064146

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