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Properties

Label 2-1800-24.11-c1-0-37
Degree $2$
Conductor $1800$
Sign $0.954 + 0.297i$
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  + (−1.38 + 0.305i)2-s + (1.81 − 0.844i)4-s − 1.41i·7-s + (−2.24 + 1.71i)8-s + 0.191i·11-s − 2.63i·13-s + (0.432 + 1.95i)14-s + (2.57 − 3.06i)16-s + 6.20i·17-s − 1.52·19-s + (−0.0585 − 0.264i)22-s + 5.25·23-s + (0.806 + 3.64i)26-s + (−1.19 − 2.56i)28-s − 0.270·29-s + ⋯
L(s)  = 1  + (−0.976 + 0.216i)2-s + (0.906 − 0.422i)4-s − 0.534i·7-s + (−0.793 + 0.608i)8-s + 0.0577i·11-s − 0.731i·13-s + (0.115 + 0.521i)14-s + (0.643 − 0.765i)16-s + 1.50i·17-s − 0.349·19-s + (−0.0124 − 0.0563i)22-s + 1.09·23-s + (0.158 + 0.714i)26-s + (−0.225 − 0.484i)28-s − 0.0502·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.954 + 0.297i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.058716574\)
\(L(\frac12)\) \(\approx\) \(1.058716574\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.305i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 - 0.191iT - 11T^{2} \)
13 \( 1 + 2.63iT - 13T^{2} \)
17 \( 1 - 6.20iT - 17T^{2} \)
19 \( 1 + 1.52T + 19T^{2} \)
23 \( 1 - 5.25T + 23T^{2} \)
29 \( 1 + 0.270T + 29T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 + 7.61iT - 37T^{2} \)
41 \( 1 + 9.22iT - 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + 3.79T + 47T^{2} \)
53 \( 1 - 8.77T + 53T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 + 0.382iT - 61T^{2} \)
67 \( 1 + 1.72T + 67T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 - 5.45T + 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + 15.2iT - 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 + 7.31T + 97T^{2} \)
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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

−8.906208235403378820946964114598, −8.714305705600110987858196407511, −7.54081616340834718696273838092, −7.20230832698193488416180677730, −6.10947526156611301792449766726, −5.49724362420598410412853268554, −4.20072494481432589299795844667, −3.12704117521793972498201720410, −1.93302806043394279273300634174, −0.73332656158974230656505223619, 0.913408531030874371717806414364, 2.28394101357306519360971580781, 2.98919996431091715463861459722, 4.27375885373632938695910722368, 5.36009155388023317818321265896, 6.38934339636538907592208870346, 7.04516604477080666824923618472, 7.85099736576917650745274683535, 8.681480567114023326118374284944, 9.372034202460861425007610273112

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