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Properties

Label 2-30e2-9.2-c2-0-21
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  + (1.83 − 2.37i)3-s + (−4.13 + 7.15i)7-s + (−2.23 − 8.71i)9-s + (−1.19 − 0.687i)11-s + (10.7 + 18.6i)13-s − 25.3i·17-s − 2.49·19-s + (9.36 + 22.9i)21-s + (33.1 − 19.1i)23-s + (−24.7 − 10.7i)27-s + (37.0 + 21.4i)29-s + (10.9 + 18.9i)31-s + (−3.81 + 1.55i)33-s + 30.5·37-s + (64.0 + 8.78i)39-s + ⋯
L(s)  = 1  + (0.612 − 0.790i)3-s + (−0.590 + 1.02i)7-s + (−0.248 − 0.968i)9-s + (−0.108 − 0.0624i)11-s + (0.829 + 1.43i)13-s − 1.48i·17-s − 0.131·19-s + (0.445 + 1.09i)21-s + (1.44 − 0.833i)23-s + (−0.917 − 0.397i)27-s + (1.27 + 0.738i)29-s + (0.353 + 0.611i)31-s + (−0.115 + 0.0472i)33-s + 0.826·37-s + (1.64 + 0.225i)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: $\chi_{900} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.901 + 0.432i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.290531330\)
\(L(\frac12)\) \(\approx\) \(2.290531330\)
\(L(2)\) 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 \( 1 + (-1.83 + 2.37i)T \)
5 \( 1 \)
good7 \( 1 + (4.13 - 7.15i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (1.19 + 0.687i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.7 - 18.6i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 25.3iT - 289T^{2} \)
19 \( 1 + 2.49T + 361T^{2} \)
23 \( 1 + (-33.1 + 19.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-37.0 - 21.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-10.9 - 18.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 30.5T + 1.36e3T^{2} \)
41 \( 1 + (-7.29 + 4.21i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-35.5 + 61.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-42.1 - 24.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 1.96iT - 2.80e3T^{2} \)
59 \( 1 + (-3.77 + 2.18i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (18.4 - 32.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.06 + 13.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 71.5iT - 5.04e3T^{2} \)
73 \( 1 - 122.T + 5.32e3T^{2} \)
79 \( 1 + (-3.98 + 6.90i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (90.2 + 52.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 9.37iT - 7.92e3T^{2} \)
97 \( 1 + (6.86 - 11.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.393517562710714218300793388830, −9.021198909728190798550693321685, −8.397865770309912504585023679105, −7.05137623940858706647938465344, −6.68428570888187380413385910860, −5.67019908760514549486773688558, −4.43388230345152828105409881261, −3.05902275949519728654805925976, −2.41712074339012642251313944626, −0.964187191071330229251618929039, 0.970973498607528176143474447452, 2.77489187984082403438773287374, 3.61721461369763446230582688018, 4.36794667059922651051122484481, 5.56996739908629218162329263674, 6.48482545433777408580817881615, 7.76013794303070771227282630445, 8.193170298496570793590533366004, 9.246108212627111394803489862479, 10.06004771658613568968297610480

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