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Properties

Label 2-30e2-9.5-c2-0-18
Degree $2$
Conductor $900$
Sign $0.689 + 0.724i$
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  + (−0.0930 − 2.99i)3-s + (3.67 + 6.36i)7-s + (−8.98 + 0.557i)9-s + (−8.35 + 4.82i)11-s + (8.34 − 14.4i)13-s + 15.4i·17-s + 17.4·19-s + (18.7 − 11.6i)21-s + (−9.16 − 5.29i)23-s + (2.50 + 26.8i)27-s + (36.2 − 20.9i)29-s + (24.3 − 42.1i)31-s + (15.2 + 24.6i)33-s + 20.4·37-s + (−44.1 − 23.6i)39-s + ⋯
L(s)  = 1  + (−0.0310 − 0.999i)3-s + (0.524 + 0.908i)7-s + (−0.998 + 0.0619i)9-s + (−0.759 + 0.438i)11-s + (0.641 − 1.11i)13-s + 0.910i·17-s + 0.920·19-s + (0.892 − 0.552i)21-s + (−0.398 − 0.230i)23-s + (0.0928 + 0.995i)27-s + (1.24 − 0.720i)29-s + (0.785 − 1.36i)31-s + (0.461 + 0.745i)33-s + 0.553·37-s + (−1.13 − 0.606i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.689 + 0.724i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.689 + 0.724i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.887955139\)
\(L(\frac12)\) \(\approx\) \(1.887955139\)
\(L(2)\) 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 \( 1 + (0.0930 + 2.99i)T \)
5 \( 1 \)
good7 \( 1 + (-3.67 - 6.36i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.35 - 4.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.34 + 14.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 15.4iT - 289T^{2} \)
19 \( 1 - 17.4T + 361T^{2} \)
23 \( 1 + (9.16 + 5.29i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-36.2 + 20.9i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-24.3 + 42.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 20.4T + 1.36e3T^{2} \)
41 \( 1 + (-24.1 - 13.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-15.1 - 26.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.4 + 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 90.4iT - 2.80e3T^{2} \)
59 \( 1 + (-48.0 - 27.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (58.3 + 101. i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (8.64 - 14.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 38.4iT - 5.04e3T^{2} \)
73 \( 1 + 7.38T + 5.32e3T^{2} \)
79 \( 1 + (-11.8 - 20.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-3.69 + 2.13i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + (51.3 + 88.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.829979953114625886309842982794, −8.645737916535574264236224191502, −8.040988841321191081048723003028, −7.52346513304216305964517061736, −6.05718752531256691548050366789, −5.80627568579252019030896450912, −4.59599472070218303719812739776, −2.99841879558303185841807878143, −2.18028797932429949081826795490, −0.839726129512031823277522833008, 0.960309119978430409775844274060, 2.74517035085337966122709242177, 3.79024242981534862914664775363, 4.67198621259116535218545169029, 5.39608764230614674045649321653, 6.58075194947358535350779847581, 7.55729381746182015518046140617, 8.488431385124421091957212614487, 9.214329691558780670937796772559, 10.16123499167535912546877283571

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