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Properties

Label 2-1800-8.5-c1-0-78
Degree $2$
Conductor $1800$
Sign $0.825 + 0.563i$
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  + (0.450 + 1.34i)2-s + (−1.59 + 1.20i)4-s + 2.64·7-s + (−2.33 − 1.59i)8-s − 1.51i·11-s − 3.87i·13-s + (1.18 + 3.54i)14-s + (1.08 − 3.84i)16-s − 3.31·17-s − 7.08i·19-s + (2.02 − 0.681i)22-s − 4.82·23-s + (5.18 − 1.74i)26-s + (−4.21 + 3.18i)28-s − 2.18i·29-s + ⋯
L(s)  = 1  + (0.318 + 0.947i)2-s + (−0.797 + 0.603i)4-s + 0.998·7-s + (−0.825 − 0.563i)8-s − 0.456i·11-s − 1.07i·13-s + (0.317 + 0.946i)14-s + (0.271 − 0.962i)16-s − 0.803·17-s − 1.62i·19-s + (0.432 − 0.145i)22-s − 1.00·23-s + (1.01 − 0.341i)26-s + (−0.796 + 0.602i)28-s − 0.405i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.825 + 0.563i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.825 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.325077635\)
\(L(\frac12)\) \(\approx\) \(1.325077635\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.450 - 1.34i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 + 1.51iT - 11T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 + 7.08iT - 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 2.18iT - 29T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 + 7.87iT - 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 - 7.08T + 47T^{2} \)
53 \( 1 - 4.50iT - 53T^{2} \)
59 \( 1 + 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 + 1.01iT - 67T^{2} \)
71 \( 1 - 6.72T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−8.862517710697911234495348978089, −8.350501149148226197174811851689, −7.57506486904166959461335341525, −6.90222429121196097490457429902, −5.87059417545634205587315365877, −5.23330592798358228458003215514, −4.45531688724562805659643114350, −3.50548559384548748291306559251, −2.28002298667307138514262668040, −0.43197090410975936843594880148, 1.61111233382950095388842259583, 2.05964816526173617372743134103, 3.55970711847120210681632985459, 4.30108678352732397729547860345, 5.02494556663721221795784868266, 5.93436532149432666101875453913, 6.91465723299207127419892096384, 8.042559045567456680377737649439, 8.636089971403713515494852108405, 9.521911814935136645132683024349

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