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Properties

Label 2-1800-120.59-c1-0-26
Degree $2$
Conductor $1800$
Sign $0.999 + 0.0381i$
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  + (−1.25 − 0.660i)2-s + (1.12 + 1.65i)4-s + 1.68·7-s + (−0.320 − 2.81i)8-s + 2.05i·11-s + 0.247·13-s + (−2.10 − 1.11i)14-s + (−1.45 + 3.72i)16-s − 1.13·17-s + 4.79·19-s + (1.35 − 2.57i)22-s − 5.26i·23-s + (−0.309 − 0.163i)26-s + (1.89 + 2.77i)28-s − 0.599·29-s + ⋯
L(s)  = 1  + (−0.884 − 0.466i)2-s + (0.564 + 0.825i)4-s + 0.636·7-s + (−0.113 − 0.993i)8-s + 0.619i·11-s + 0.0687·13-s + (−0.562 − 0.296i)14-s + (−0.363 + 0.931i)16-s − 0.275·17-s + 1.09·19-s + (0.289 − 0.548i)22-s − 1.09i·23-s + (−0.0607 − 0.0320i)26-s + (0.358 + 0.525i)28-s − 0.111·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0381i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.999 + 0.0381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.235488122\)
\(L(\frac12)\) \(\approx\) \(1.235488122\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.660i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.68T + 7T^{2} \)
11 \( 1 - 2.05iT - 11T^{2} \)
13 \( 1 - 0.247T + 13T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 4.79T + 19T^{2} \)
23 \( 1 + 5.26iT - 23T^{2} \)
29 \( 1 + 0.599T + 29T^{2} \)
31 \( 1 - 6.26iT - 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 - 3.83iT - 41T^{2} \)
43 \( 1 + 3.88iT - 43T^{2} \)
47 \( 1 - 6.96iT - 47T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 + 0.397iT - 59T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 - 0.232iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 7.98iT - 73T^{2} \)
79 \( 1 + 10.8iT - 79T^{2} \)
83 \( 1 + 3.22T + 83T^{2} \)
89 \( 1 - 16.3iT - 89T^{2} \)
97 \( 1 - 16.4iT - 97T^{2} \)
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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

−9.393322250840055154542322618470, −8.461903862705727304641264953422, −7.899269870479066900967241299144, −7.09937628199180576430457157859, −6.33765627805733084169412131335, −5.06879578032913194065581529804, −4.21196398349958627310919090972, −3.06426098399500593353912772767, −2.09843015136791427512374705149, −0.986202341884660845706584488026, 0.813823923928121663012060151210, 1.96420201566478323300060063261, 3.17534687844119605100392566233, 4.50553089850538922688593934149, 5.52108289507028189773502461972, 6.05893209985386089543553852514, 7.15390442915615467981272585530, 7.79469519002796694851862765229, 8.372605361263041448147330543540, 9.351884694640240987713493218182

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