Properties

Label 2-1800-8.5-c1-0-30
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

\(L(1)\) \(\approx\) \(1.998695977\)
\(L(\frac12)\) \(\approx\) \(1.998695977\)
\(L(\frac{3}{2})\) 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

−9.251956166203839980078685735694, −8.955757051518340124362863211992, −7.39389428353887153187992289728, −6.88773199319206074744192124046, −5.86076670221262715971274276672, −4.85487528008498155119689290777, −4.36412849263797101292094622818, −3.23583798925990949942502778059, −2.30787486215559479927093044459, −1.22572114099156985419083186889, 0.69849182332514504161715399679, 2.86014267371486368250676389943, 3.36272665139321949433189804206, 4.45533169120703348974449859219, 5.49384048734057389665599177302, 5.98493182464189907680103581857, 6.76866788729576619167533833111, 7.85189956077013771129012703076, 8.221202035198546409861692705750, 9.115726452449161618122028401331

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