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Properties

Label 2-30e2-45.29-c2-0-7
Degree $2$
Conductor $900$
Sign $0.171 - 0.985i$
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  + (−2.05 + 2.18i)3-s + (−7.02 − 4.05i)7-s + (−0.558 − 8.98i)9-s + (−17.6 − 10.1i)11-s + (5.29 − 3.05i)13-s + 17.9·17-s − 9.11·19-s + (23.3 − 7.02i)21-s + (16.7 + 29.0i)23-s + (20.7 + 17.2i)27-s + (−14.4 − 8.31i)29-s + (11.1 + 19.3i)31-s + (58.4 − 17.6i)33-s + 50.4i·37-s + (−4.19 + 17.8i)39-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)3-s + (−1.00 − 0.579i)7-s + (−0.0620 − 0.998i)9-s + (−1.60 − 0.924i)11-s + (0.407 − 0.235i)13-s + 1.05·17-s − 0.479·19-s + (1.11 − 0.334i)21-s + (0.729 + 1.26i)23-s + (0.769 + 0.638i)27-s + (−0.496 − 0.286i)29-s + (0.360 + 0.624i)31-s + (1.77 − 0.533i)33-s + 1.36i·37-s + (−0.107 + 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.171 - 0.985i$
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: \((2,\ 900,\ (\ :1),\ 0.171 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7421964313\)
\(L(\frac12)\) \(\approx\) \(0.7421964313\)
\(L(2)\) 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 \( 1 + (2.05 - 2.18i)T \)
5 \( 1 \)
good7 \( 1 + (7.02 + 4.05i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (17.6 + 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.29 + 3.05i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 17.9T + 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (-16.7 - 29.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (14.4 + 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 50.4iT - 1.36e3T^{2} \)
41 \( 1 + (-29.9 + 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.9 + 11.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (19.1 - 33.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 19.0T + 2.80e3T^{2} \)
59 \( 1 + (-2.96 + 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.45 + 3.14i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 47.3iT - 5.32e3T^{2} \)
79 \( 1 + (42.2 - 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-19.1 + 33.1i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (-69.9 - 40.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.15479180379052213390783516678, −9.572841796945138269708399206431, −8.459250358558234651332674539480, −7.54883874497864019739413267746, −6.46461819795350319294256929684, −5.67617902746905491552189721031, −4.96934895967197777974731452138, −3.58593099893320078981380348802, −3.07749028896389917170516973700, −0.822324429850365940067472494495, 0.36930291981909113882719171689, 2.07881307322264645270763058692, 2.98191191907606692152093732800, 4.58701255620731674673937784679, 5.52771965495774545010936160356, 6.21425315482166272537746072429, 7.14867597281798086163828895147, 7.85141548868272535754631748542, 8.831666329030575105311913542788, 9.942533357168211234748986958133

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