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Properties

Label 2-30e2-9.5-c2-0-0
Degree $2$
Conductor $900$
Sign $-0.152 - 0.988i$
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  + (−2.88 − 0.807i)3-s + (−6.19 − 10.7i)7-s + (7.69 + 4.66i)9-s + (−4.00 + 2.31i)11-s + (−0.631 + 1.09i)13-s − 22.9i·17-s − 32.8·19-s + (9.22 + 35.9i)21-s + (−0.690 − 0.398i)23-s + (−18.4 − 19.6i)27-s + (34.3 − 19.8i)29-s + (−0.730 + 1.26i)31-s + (13.4 − 3.44i)33-s + 44.9·37-s + (2.70 − 2.65i)39-s + ⋯
L(s)  = 1  + (−0.963 − 0.269i)3-s + (−0.884 − 1.53i)7-s + (0.855 + 0.518i)9-s + (−0.364 + 0.210i)11-s + (−0.0485 + 0.0841i)13-s − 1.34i·17-s − 1.73·19-s + (0.439 + 1.71i)21-s + (−0.0300 − 0.0173i)23-s + (−0.683 − 0.729i)27-s + (1.18 − 0.683i)29-s + (−0.0235 + 0.0407i)31-s + (0.407 − 0.104i)33-s + 1.21·37-s + (0.0694 − 0.0679i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.152 - 0.988i$
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: \((2,\ 900,\ (\ :1),\ -0.152 - 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1170886023\)
\(L(\frac12)\) \(\approx\) \(0.1170886023\)
\(L(2)\) 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 \( 1 + (2.88 + 0.807i)T \)
5 \( 1 \)
good7 \( 1 + (6.19 + 10.7i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.00 - 2.31i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (0.631 - 1.09i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 22.9iT - 289T^{2} \)
19 \( 1 + 32.8T + 361T^{2} \)
23 \( 1 + (0.690 + 0.398i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-34.3 + 19.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (0.730 - 1.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 44.9T + 1.36e3T^{2} \)
41 \( 1 + (33.8 + 19.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-18.1 - 31.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (52.8 - 30.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 13.9iT - 2.80e3T^{2} \)
59 \( 1 + (-50.2 - 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-27.5 - 47.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (33.4 - 58.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 70.6iT - 5.04e3T^{2} \)
73 \( 1 + 23.1T + 5.32e3T^{2} \)
79 \( 1 + (14.5 + 25.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (112. - 65.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 48.0iT - 7.92e3T^{2} \)
97 \( 1 + (9.68 + 16.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.14071095783944048331096420012, −9.734599013362779950596299447506, −8.296336005294607477587633397623, −7.27835689886780466335461182323, −6.78486392995789201678633575575, −6.00385185620957330939461826711, −4.69941656439051375290419161889, −4.12270880089850626684672357003, −2.63116528497241990541161208189, −0.946677891075445657486543637435, 0.05351953988739473981621089696, 1.97357244600857641721389171666, 3.23524575678354023217941901418, 4.43695570763709811783605090251, 5.46708760469949273022639405050, 6.20688486345523924217502587060, 6.65545502339886820222960061522, 8.234536258478866446329109316675, 8.837186239879967024138961702267, 9.856494602413543573383956560597

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