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Properties

Label 2-30e2-5.3-c2-0-11
Degree $2$
Conductor $900$
Sign $-0.850 + 0.525i$
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  + (−7.44 + 7.44i)7-s + 16.2·11-s + (−12.2 − 12.2i)13-s + (−7.55 + 7.55i)17-s + 14.4i·19-s + (−2.65 − 2.65i)23-s − 34.2i·29-s − 20.4·31-s + (−7.34 + 7.34i)37-s − 25.5·41-s + (−25.1 − 25.1i)43-s + (22.0 − 22.0i)47-s − 61.9i·49-s + (−35.3 − 35.3i)53-s − 88.7i·59-s + ⋯
L(s)  = 1  + (−1.06 + 1.06i)7-s + 1.47·11-s + (−0.942 − 0.942i)13-s + (−0.444 + 0.444i)17-s + 0.762i·19-s + (−0.115 − 0.115i)23-s − 1.18i·29-s − 0.661·31-s + (−0.198 + 0.198i)37-s − 0.622·41-s + (−0.583 − 0.583i)43-s + (0.469 − 0.469i)47-s − 1.26i·49-s + (−0.666 − 0.666i)53-s − 1.50i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1819385256\)
\(L(\frac12)\) \(\approx\) \(0.1819385256\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (7.44 - 7.44i)T - 49iT^{2} \)
11 \( 1 - 16.2T + 121T^{2} \)
13 \( 1 + (12.2 + 12.2i)T + 169iT^{2} \)
17 \( 1 + (7.55 - 7.55i)T - 289iT^{2} \)
19 \( 1 - 14.4iT - 361T^{2} \)
23 \( 1 + (2.65 + 2.65i)T + 529iT^{2} \)
29 \( 1 + 34.2iT - 841T^{2} \)
31 \( 1 + 20.4T + 961T^{2} \)
37 \( 1 + (7.34 - 7.34i)T - 1.36e3iT^{2} \)
41 \( 1 + 25.5T + 1.68e3T^{2} \)
43 \( 1 + (25.1 + 25.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-22.0 + 22.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (35.3 + 35.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 88.7iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + (-24.6 + 24.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 77.9T + 5.04e3T^{2} \)
73 \( 1 + (-44.1 - 44.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 48.4iT - 6.24e3T^{2} \)
83 \( 1 + (101. + 101. i)T + 6.88e3iT^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + (55.4 - 55.4i)T - 9.40e3iT^{2} \)
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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

−9.596151715695344837452411513515, −8.847975338992553793440691412272, −7.981504521585240632276171933677, −6.79711186214701889071575957603, −6.15575003248726827901780024662, −5.31968771731182756484833953663, −3.99709505015392392678793298047, −3.07211526923617230070337120269, −1.89806216474052496092223482801, −0.05827949042895518755581257476, 1.43141534876750915223718995505, 2.98123663213831171756239885986, 3.99792969105739034832541578476, 4.74065935577746486067368757365, 6.21175864535319024808327237093, 6.96530252154549201026880011795, 7.32184237033200380430527161811, 8.969053136102489065444362366863, 9.309372584910599298274980811777, 10.15474425574544095009755653348

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