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Properties

Label 2-30e2-45.29-c2-0-11
Degree $2$
Conductor $900$
Sign $0.901 - 0.432i$
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  + (−1.28 + 2.70i)3-s + (−9.36 − 5.40i)7-s + (−5.67 − 6.98i)9-s + (7.78 + 4.49i)11-s + (−3.96 + 2.28i)13-s − 7.55·17-s − 8.73·19-s + (26.7 − 18.3i)21-s + (−4.02 − 6.96i)23-s + (26.2 − 6.36i)27-s + (38.2 + 22.0i)29-s + (4.53 + 7.85i)31-s + (−22.2 + 15.2i)33-s − 56.1i·37-s + (−1.08 − 13.6i)39-s + ⋯
L(s)  = 1  + (−0.429 + 0.902i)3-s + (−1.33 − 0.772i)7-s + (−0.630 − 0.776i)9-s + (0.708 + 0.408i)11-s + (−0.304 + 0.175i)13-s − 0.444·17-s − 0.459·19-s + (1.27 − 0.875i)21-s + (−0.174 − 0.302i)23-s + (0.971 − 0.235i)27-s + (1.31 + 0.760i)29-s + (0.146 + 0.253i)31-s + (−0.673 + 0.463i)33-s − 1.51i·37-s + (−0.0278 − 0.350i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.901 - 0.432i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.901 - 0.432i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.110737781\)
\(L(\frac12)\) \(\approx\) \(1.110737781\)
\(L(2)\) 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 \( 1 + (1.28 - 2.70i)T \)
5 \( 1 \)
good7 \( 1 + (9.36 + 5.40i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.78 - 4.49i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.96 - 2.28i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 7.55T + 289T^{2} \)
19 \( 1 + 8.73T + 361T^{2} \)
23 \( 1 + (4.02 + 6.96i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-38.2 - 22.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4.53 - 7.85i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 56.1iT - 1.36e3T^{2} \)
41 \( 1 + (-53.9 + 31.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-34.7 - 20.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.10 - 15.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 23.7T + 2.80e3T^{2} \)
59 \( 1 + (-59.3 + 34.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-16.3 + 28.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.20 + 4.15i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 115. iT - 5.04e3T^{2} \)
73 \( 1 - 125. iT - 5.32e3T^{2} \)
79 \( 1 + (15.5 - 26.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (77.0 - 133. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-143. - 83.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.965904642505250050562250754273, −9.368558821191310162863421288365, −8.582986599473821487997494582351, −7.08104021917036765550300616233, −6.59310233091671736572558320547, −5.64152387451182545466247303508, −4.36328253561157684129323786115, −3.87990486235529419089347559441, −2.68267492095321608085440992670, −0.64572164341905618549932423818, 0.68704905024295123236013014960, 2.26912208839721047056994536442, 3.17158283835634697331761482576, 4.59579188939548109381506142980, 5.95425147343175386074272301019, 6.24858216784621913013395285342, 7.10071283802081488235890374396, 8.193268776098031110174714384963, 8.958020937960098076937234264497, 9.813724581573197713333486541781

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