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Properties

Label 2-1800-8.5-c1-0-48
Degree $2$
Conductor $1800$
Sign $0.979 - 0.201i$
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  + (−0.622 + 1.26i)2-s + (−1.22 − 1.58i)4-s + 1.44·7-s + (2.77 − 0.570i)8-s + 1.14i·11-s − 3.87i·13-s + (−0.902 + 1.84i)14-s + (−0.999 + 3.87i)16-s − 3.05·17-s − 0.710i·19-s + (−1.44 − 0.710i)22-s + 5.54·23-s + (4.91 + 2.41i)26-s + (−1.77 − 2.29i)28-s + 6.22i·29-s + ⋯
L(s)  = 1  + (−0.440 + 0.897i)2-s + (−0.612 − 0.790i)4-s + 0.547·7-s + (0.979 − 0.201i)8-s + 0.344i·11-s − 1.07i·13-s + (−0.241 + 0.491i)14-s + (−0.249 + 0.968i)16-s − 0.739·17-s − 0.163i·19-s + (−0.309 − 0.151i)22-s + 1.15·23-s + (0.964 + 0.472i)26-s + (−0.335 − 0.433i)28-s + 1.15i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.979 - 0.201i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.979 - 0.201i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.286409255\)
\(L(\frac12)\) \(\approx\) \(1.286409255\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.622 - 1.26i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 - 1.14iT - 11T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 + 0.710iT - 19T^{2} \)
23 \( 1 - 5.54T + 23T^{2} \)
29 \( 1 - 6.22iT - 29T^{2} \)
31 \( 1 - 3.44T + 31T^{2} \)
37 \( 1 + 6.32iT - 37T^{2} \)
41 \( 1 - 8.03T + 41T^{2} \)
43 \( 1 + 7.03iT - 43T^{2} \)
47 \( 1 + 4.98T + 47T^{2} \)
53 \( 1 + 3.93iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 + 2.45iT - 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−9.137578910403132972758847795673, −8.435544107013033768222954332967, −7.73189535495535276814456776510, −7.01476596349359722257721830088, −6.23946860894074451003214108638, −5.17445725759262109818913827791, −4.79124248911931903107317568605, −3.50610653305989974831247170487, −2.06071159504008873876306112432, −0.69770601434804257200293022470, 1.05457960485009920290374347974, 2.15730334899564456653648152212, 3.10300854354423157112961738556, 4.31520452353977615721451301276, 4.74248585346888785137775832446, 6.08776643885348285834791067586, 7.04604092754577013190656146076, 7.921709719835037479903936603524, 8.631042520222771384562723846402, 9.269510656777484700802032855994

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