Properties

Label 2-30e2-5.2-c4-0-15
Degree $2$
Conductor $900$
Sign $0.850 + 0.525i$
Analytic cond. $93.0329$
Root an. cond. $9.64535$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−50.7 − 50.7i)7-s − 2.62·11-s + (43.4 − 43.4i)13-s + (131. + 131. i)17-s + 403. i·19-s + (−334. + 334. i)23-s + 1.17e3i·29-s + 955.·31-s + (−673. − 673. i)37-s − 818.·41-s + (−2.48 + 2.48i)43-s + (−1.56e3 − 1.56e3i)47-s + 2.75e3i·49-s + (277. − 277. i)53-s − 6.33e3i·59-s + ⋯
L(s)  = 1  + (−1.03 − 1.03i)7-s − 0.0216·11-s + (0.256 − 0.256i)13-s + (0.454 + 0.454i)17-s + 1.11i·19-s + (−0.632 + 0.632i)23-s + 1.39i·29-s + 0.994·31-s + (−0.491 − 0.491i)37-s − 0.486·41-s + (−0.00134 + 0.00134i)43-s + (−0.707 − 0.707i)47-s + 1.14i·49-s + (0.0988 − 0.0988i)53-s − 1.81i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(93.0329\)
Root analytic conductor: \(9.64535\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :2),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.572257656\)
\(L(\frac12)\) \(\approx\) \(1.572257656\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (50.7 + 50.7i)T + 2.40e3iT^{2} \)
11 \( 1 + 2.62T + 1.46e4T^{2} \)
13 \( 1 + (-43.4 + 43.4i)T - 2.85e4iT^{2} \)
17 \( 1 + (-131. - 131. i)T + 8.35e4iT^{2} \)
19 \( 1 - 403. iT - 1.30e5T^{2} \)
23 \( 1 + (334. - 334. i)T - 2.79e5iT^{2} \)
29 \( 1 - 1.17e3iT - 7.07e5T^{2} \)
31 \( 1 - 955.T + 9.23e5T^{2} \)
37 \( 1 + (673. + 673. i)T + 1.87e6iT^{2} \)
41 \( 1 + 818.T + 2.82e6T^{2} \)
43 \( 1 + (2.48 - 2.48i)T - 3.41e6iT^{2} \)
47 \( 1 + (1.56e3 + 1.56e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-277. + 277. i)T - 7.89e6iT^{2} \)
59 \( 1 + 6.33e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.51e3T + 1.38e7T^{2} \)
67 \( 1 + (-713. - 713. i)T + 2.01e7iT^{2} \)
71 \( 1 + 288.T + 2.54e7T^{2} \)
73 \( 1 + (-5.56e3 + 5.56e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 4.06e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.28e3 + 1.28e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 4.41e3iT - 6.27e7T^{2} \)
97 \( 1 + (-8.80e3 - 8.80e3i)T + 8.85e7iT^{2} \)
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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

−9.760049293907960315338468084006, −8.555200401003190867529047745007, −7.76210111359268725017419454851, −6.86063910007898688377845674150, −6.13412218396944578806784083602, −5.10773502156321824997257017399, −3.74705829581344701155825063530, −3.38073680810637320867162570300, −1.73670217307441958113420926266, −0.54540205381973245190225483666, 0.66795910841629601522030117655, 2.31169570030162944127147371149, 3.02963439243155358704268441794, 4.24759520088696609014196176523, 5.34204678313239462983018985799, 6.23703889984116267063478227528, 6.85319154952165122071774661632, 8.076685095406774153274854734077, 8.840171691039822307718631843487, 9.628147866692677444286280282631

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