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Properties

Label 2-1800-24.11-c1-0-7
Degree $2$
Conductor $1800$
Sign $-0.995 + 0.0932i$
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  + (0.331 + 1.37i)2-s + (−1.78 + 0.910i)4-s − 0.936i·7-s + (−1.84 − 2.14i)8-s + 2.20i·11-s + 3.33i·13-s + (1.28 − 0.310i)14-s + (2.34 − 3.24i)16-s + 1.54i·17-s + 3.12·19-s + (−3.03 + 0.731i)22-s + 3.39·23-s + (−4.58 + 1.10i)26-s + (0.852 + 1.66i)28-s − 8.44·29-s + ⋯
L(s)  = 1  + (0.234 + 0.972i)2-s + (−0.890 + 0.455i)4-s − 0.353i·7-s + (−0.650 − 0.759i)8-s + 0.665i·11-s + 0.924i·13-s + (0.344 − 0.0828i)14-s + (0.585 − 0.810i)16-s + 0.374i·17-s + 0.716·19-s + (−0.647 + 0.155i)22-s + 0.707·23-s + (−0.899 + 0.216i)26-s + (0.161 + 0.315i)28-s − 1.56·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.995 + 0.0932i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -0.995 + 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.086249706\)
\(L(\frac12)\) \(\approx\) \(1.086249706\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.331 - 1.37i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.936iT - 7T^{2} \)
11 \( 1 - 2.20iT - 11T^{2} \)
13 \( 1 - 3.33iT - 13T^{2} \)
17 \( 1 - 1.54iT - 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 + 8.44T + 29T^{2} \)
31 \( 1 - 8.30iT - 31T^{2} \)
37 \( 1 + 7.60iT - 37T^{2} \)
41 \( 1 - 5.83iT - 41T^{2} \)
43 \( 1 + 7.77T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 5.59T + 73T^{2} \)
79 \( 1 + 1.02iT - 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 - 13.0iT - 89T^{2} \)
97 \( 1 + 2.18T + 97T^{2} \)
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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

−9.409535184229022672005463538317, −8.897654646092229554308438684031, −7.907524272692106608043696568811, −7.17278121183128679413510411201, −6.69901781839934536944647312132, −5.65665553853256327679757348178, −4.86347825366081987254037729325, −4.08038216613862578419336473825, −3.15719095999260024207872208649, −1.53386903590693862384342872130, 0.38135812822674237884447352444, 1.73136563489119390576809121809, 2.96464610805103479503287057296, 3.49430589522518187771451236147, 4.74097872321008043521855241703, 5.47138048031824498220011467905, 6.14925661441136367558718500217, 7.46579539748514203318307763696, 8.269354675811050327084210409207, 9.059326405610369230166555173581

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