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Properties

Label 2-1800-40.29-c1-0-51
Degree $2$
Conductor $1800$
Sign $0.856 + 0.516i$
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  + (−1.38 + 0.264i)2-s + (1.85 − 0.735i)4-s − 0.941i·7-s + (−2.38 + 1.51i)8-s − 4.49i·11-s + 5.55·13-s + (0.249 + 1.30i)14-s + (2.91 − 2.73i)16-s + 7.55i·17-s − 1.05i·19-s + (1.19 + 6.24i)22-s + 1.05i·23-s + (−7.71 + 1.47i)26-s + (−0.692 − 1.75i)28-s − 2i·29-s + ⋯
L(s)  = 1  + (−0.982 + 0.187i)2-s + (0.929 − 0.367i)4-s − 0.355i·7-s + (−0.844 + 0.535i)8-s − 1.35i·11-s + 1.54·13-s + (0.0665 + 0.349i)14-s + (0.729 − 0.683i)16-s + 1.83i·17-s − 0.242i·19-s + (0.253 + 1.33i)22-s + 0.220i·23-s + (−1.51 + 0.288i)26-s + (−0.130 − 0.330i)28-s − 0.371i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.856 + 0.516i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.151577894\)
\(L(\frac12)\) \(\approx\) \(1.151577894\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.264i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.941iT - 7T^{2} \)
11 \( 1 + 4.49iT - 11T^{2} \)
13 \( 1 - 5.55T + 13T^{2} \)
17 \( 1 - 7.55iT - 17T^{2} \)
19 \( 1 + 1.05iT - 19T^{2} \)
23 \( 1 - 1.05iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 3.55T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 - 3.88T + 41T^{2} \)
43 \( 1 - 1.88T + 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 8.49iT - 59T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 5.88T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−8.839411621758142226960275033695, −8.569421821241348767523007135240, −7.907427970531853547784823022590, −6.82702172558528317957916039008, −6.09440419312783427574271655617, −5.60820037623355783065350792208, −3.96512288430290512516310952949, −3.23973054348349546009941508874, −1.79751340944718642425032751558, −0.74993465261889084267715236367, 1.03780618642727452991733022835, 2.19620152091604425398711265869, 3.13408107313426940501268193192, 4.28962958373711818799458828645, 5.41472602671618974989616613382, 6.41446323615046007187530436434, 7.12001578470614585066849696365, 7.81810700473469176570408474584, 8.751475651232918299865848988234, 9.271943554754570363196389320762

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