Properties

Label 2-30e2-20.19-c2-0-67
Degree $2$
Conductor $900$
Sign $0.922 + 0.385i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.922 + 0.385i$
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: \((2,\ 900,\ (\ :1),\ 0.922 + 0.385i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.552887411\)
\(L(\frac12)\) \(\approx\) \(1.552887411\)
\(L(2)\) 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

−9.380620536628659218822839917262, −9.034186546953054054397902382600, −7.955258819355531445924822344520, −7.37042590825254962008061006821, −6.37888474328954080731467907041, −5.51082466040578691074028134644, −4.66775220740122549729852836956, −3.76537820862557536744962584254, −2.42027570267946773004746704333, −0.46027560854055808382078930367, 1.42105954312466276784817795128, 2.23658791972678349119278391019, 3.71144120383842610506650862551, 4.36809567455548036750251091784, 5.48324676535883296279812044092, 6.16927828929545303156986127534, 7.76322513119049312352361515465, 8.234993688987878234093304983277, 9.394623896765889115697387692811, 10.23026317944003887488619118166

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