Properties

Label 2-30e2-20.19-c2-0-61
Degree $2$
Conductor $900$
Sign $-0.245 + 0.969i$
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  + (−0.696 + 1.87i)2-s + (−3.02 − 2.61i)4-s − 5.46·7-s + (7.00 − 3.86i)8-s + 11.0i·11-s + 10.1i·13-s + (3.80 − 10.2i)14-s + (2.35 + 15.8i)16-s − 24.4i·17-s + 23.7i·19-s + (−20.6 − 7.69i)22-s + 37.2·23-s + (−18.9 − 7.05i)26-s + (16.5 + 14.2i)28-s − 25.7·29-s + ⋯
L(s)  = 1  + (−0.348 + 0.937i)2-s + (−0.757 − 0.652i)4-s − 0.781·7-s + (0.875 − 0.482i)8-s + 1.00i·11-s + 0.778i·13-s + (0.272 − 0.732i)14-s + (0.147 + 0.989i)16-s − 1.43i·17-s + 1.25i·19-s + (−0.940 − 0.349i)22-s + 1.61·23-s + (−0.730 − 0.271i)26-s + (0.591 + 0.510i)28-s − 0.888·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.245 + 0.969i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01693088338\)
\(L(\frac12)\) \(\approx\) \(0.01693088338\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.696 - 1.87i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 5.46T + 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 - 10.1iT - 169T^{2} \)
17 \( 1 + 24.4iT - 289T^{2} \)
19 \( 1 - 23.7iT - 361T^{2} \)
23 \( 1 - 37.2T + 529T^{2} \)
29 \( 1 + 25.7T + 841T^{2} \)
31 \( 1 - 4.83iT - 961T^{2} \)
37 \( 1 + 35.6iT - 1.36e3T^{2} \)
41 \( 1 - 9.30T + 1.68e3T^{2} \)
43 \( 1 + 70.0T + 1.84e3T^{2} \)
47 \( 1 + 38.0T + 2.20e3T^{2} \)
53 \( 1 + 55.7iT - 2.80e3T^{2} \)
59 \( 1 + 55.5iT - 3.48e3T^{2} \)
61 \( 1 + 82.2T + 3.72e3T^{2} \)
67 \( 1 + 104.T + 4.48e3T^{2} \)
71 \( 1 - 76.7iT - 5.04e3T^{2} \)
73 \( 1 + 93.5iT - 5.32e3T^{2} \)
79 \( 1 + 49.3iT - 6.24e3T^{2} \)
83 \( 1 + 72.3T + 6.88e3T^{2} \)
89 \( 1 - 115.T + 7.92e3T^{2} \)
97 \( 1 - 72.9iT - 9.40e3T^{2} \)
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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

−9.492346765614926121685886582622, −8.994679061996194631919138370004, −7.78054303480989809625162778408, −7.06075350996262127187669438449, −6.48509109742579317821740846274, −5.33799307010199411437253250967, −4.56575500805733373193654681610, −3.34775902706731697678265277531, −1.68683072704870813401642982390, −0.00671605632877191305171607955, 1.27937828403310063930989669324, 2.89777202384451221275746461501, 3.39466370777096524429337265333, 4.65603708694866943933988531468, 5.74422901355740192186520661621, 6.79441913384687766504920088263, 7.906623383602995573092587168191, 8.711968928828442325020699917990, 9.323669173454291561609185941075, 10.30897721844371737535575514332

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