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Properties

Label 2-30e2-20.19-c2-0-45
Degree $2$
Conductor $900$
Sign $0.801 - 0.598i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 − 0.177i)2-s + (3.93 − 0.707i)4-s − 1.19·7-s + (7.71 − 2.10i)8-s + 8.22i·11-s + 11.1i·13-s + (−2.38 + 0.212i)14-s + (14.9 − 5.57i)16-s + 20.9i·17-s + 27.9i·19-s + (1.46 + 16.3i)22-s + 9.48·23-s + (1.98 + 22.2i)26-s + (−4.70 + 0.845i)28-s + 40.4·29-s + ⋯
L(s)  = 1  + (0.996 − 0.0888i)2-s + (0.984 − 0.176i)4-s − 0.170·7-s + (0.964 − 0.263i)8-s + 0.747i·11-s + 0.860i·13-s + (−0.170 + 0.0151i)14-s + (0.937 − 0.348i)16-s + 1.23i·17-s + 1.47i·19-s + (0.0663 + 0.744i)22-s + 0.412·23-s + (0.0764 + 0.857i)26-s + (−0.168 + 0.0302i)28-s + 1.39·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.690037885\)
\(L(\frac12)\) \(\approx\) \(3.690037885\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.177i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.19T + 49T^{2} \)
11 \( 1 - 8.22iT - 121T^{2} \)
13 \( 1 - 11.1iT - 169T^{2} \)
17 \( 1 - 20.9iT - 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 - 9.48T + 529T^{2} \)
29 \( 1 - 40.4T + 841T^{2} \)
31 \( 1 + 55.3iT - 961T^{2} \)
37 \( 1 + 50.1iT - 1.36e3T^{2} \)
41 \( 1 - 73.6T + 1.68e3T^{2} \)
43 \( 1 + 19.0T + 1.84e3T^{2} \)
47 \( 1 - 18.0T + 2.20e3T^{2} \)
53 \( 1 - 57.2iT - 2.80e3T^{2} \)
59 \( 1 - 60.6iT - 3.48e3T^{2} \)
61 \( 1 + 21.3T + 3.72e3T^{2} \)
67 \( 1 + 9.68T + 4.48e3T^{2} \)
71 \( 1 + 68.6iT - 5.04e3T^{2} \)
73 \( 1 - 84.7iT - 5.32e3T^{2} \)
79 \( 1 - 23.2iT - 6.24e3T^{2} \)
83 \( 1 + 93.2T + 6.88e3T^{2} \)
89 \( 1 - 62.9T + 7.92e3T^{2} \)
97 \( 1 + 91.3iT - 9.40e3T^{2} \)
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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

−10.18545317172287053813839667676, −9.335533693080539307880398207958, −8.078664795879952507582405033180, −7.31574890474089524550214280686, −6.30011936639867577681408902381, −5.73702935698485259953375866491, −4.39624785134683148801307026412, −3.93430969907817189122240746861, −2.54528159199731607546541268394, −1.51515319077821033863677055098, 0.909052258909545388650335526667, 2.75482908499168563909301640840, 3.21639967206471038723955544198, 4.71531136494113478268198946777, 5.21107102475057260730423812894, 6.38508982489503725914468445237, 6.99312049784097868838760781665, 8.006938129063336691864831634709, 8.890591076517056309487612984289, 10.02075013089887707397112246515

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