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Properties

Label 2-1800-8.5-c1-0-56
Degree $2$
Conductor $1800$
Sign $0.773 + 0.634i$
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  + (−1.37 − 0.321i)2-s + (1.79 + 0.884i)4-s + 4.05·7-s + (−2.18 − 1.79i)8-s + 0.985i·11-s − 4.94i·13-s + (−5.58 − 1.30i)14-s + (2.43 + 3.17i)16-s + 4.52·17-s + 2.60i·19-s + (0.316 − 1.35i)22-s + 3.53·23-s + (−1.58 + 6.81i)26-s + (7.27 + 3.58i)28-s − 7.59i·29-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)2-s + (0.896 + 0.442i)4-s + 1.53·7-s + (−0.773 − 0.634i)8-s + 0.297i·11-s − 1.37i·13-s + (−1.49 − 0.348i)14-s + (0.608 + 0.793i)16-s + 1.09·17-s + 0.597i·19-s + (0.0674 − 0.289i)22-s + 0.737·23-s + (−0.311 + 1.33i)26-s + (1.37 + 0.678i)28-s − 1.41i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.773 + 0.634i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.399739000\)
\(L(\frac12)\) \(\approx\) \(1.399739000\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.321i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4.05T + 7T^{2} \)
11 \( 1 - 0.985iT - 11T^{2} \)
13 \( 1 + 4.94iT - 13T^{2} \)
17 \( 1 - 4.52T + 17T^{2} \)
19 \( 1 - 2.60iT - 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + 7.59iT - 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + 0.945iT - 37T^{2} \)
41 \( 1 + 0.568T + 41T^{2} \)
43 \( 1 + 8.45iT - 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 0.229iT - 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 - 11.0iT - 61T^{2} \)
67 \( 1 + 8.45iT - 67T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 3.28T + 79T^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−9.079793681274758234791636237069, −8.348869018367438995897464620551, −7.70227953589569625117570151383, −7.33123210241640314298185020165, −5.91236666433481057450559219448, −5.30347256661694252548311128308, −4.07627683783516567940239369925, −2.96892668103635008715834068037, −1.88370810979746134961947559504, −0.868161866621204416895096429786, 1.17276088533384620651622990834, 1.95298194298965186664186169541, 3.24621095094293247230046210876, 4.67383466507458627173667673207, 5.31590364123200595683511563618, 6.37716069176878178440384915277, 7.24130801908120612925776587752, 7.79744290279773161542406536977, 8.703413149873539150365069029171, 9.092338443499633749041253644857

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