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Properties

Label 2-30e2-20.19-c2-0-44
Degree $2$
Conductor $900$
Sign $0.959 + 0.281i$
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  + (−1.99 + 0.177i)2-s + (3.93 − 0.707i)4-s + 1.19·7-s + (−7.71 + 2.10i)8-s + 8.22i·11-s − 11.1i·13-s + (−2.38 + 0.212i)14-s + (14.9 − 5.57i)16-s − 20.9i·17-s + 27.9i·19-s + (−1.46 − 16.3i)22-s − 9.48·23-s + (1.98 + 22.2i)26-s + (4.70 − 0.845i)28-s + 40.4·29-s + ⋯
L(s)  = 1  + (−0.996 + 0.0888i)2-s + (0.984 − 0.176i)4-s + 0.170·7-s + (−0.964 + 0.263i)8-s + 0.747i·11-s − 0.860i·13-s + (−0.170 + 0.0151i)14-s + (0.937 − 0.348i)16-s − 1.23i·17-s + 1.47i·19-s + (−0.0663 − 0.744i)22-s − 0.412·23-s + (0.0764 + 0.857i)26-s + (0.168 − 0.0302i)28-s + 1.39·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.959 + 0.281i$
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: \((2,\ 900,\ (\ :1),\ 0.959 + 0.281i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.148305690\)
\(L(\frac12)\) \(\approx\) \(1.148305690\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.177i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.19T + 49T^{2} \)
11 \( 1 - 8.22iT - 121T^{2} \)
13 \( 1 + 11.1iT - 169T^{2} \)
17 \( 1 + 20.9iT - 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 + 9.48T + 529T^{2} \)
29 \( 1 - 40.4T + 841T^{2} \)
31 \( 1 + 55.3iT - 961T^{2} \)
37 \( 1 - 50.1iT - 1.36e3T^{2} \)
41 \( 1 - 73.6T + 1.68e3T^{2} \)
43 \( 1 - 19.0T + 1.84e3T^{2} \)
47 \( 1 + 18.0T + 2.20e3T^{2} \)
53 \( 1 + 57.2iT - 2.80e3T^{2} \)
59 \( 1 - 60.6iT - 3.48e3T^{2} \)
61 \( 1 + 21.3T + 3.72e3T^{2} \)
67 \( 1 - 9.68T + 4.48e3T^{2} \)
71 \( 1 + 68.6iT - 5.04e3T^{2} \)
73 \( 1 + 84.7iT - 5.32e3T^{2} \)
79 \( 1 - 23.2iT - 6.24e3T^{2} \)
83 \( 1 - 93.2T + 6.88e3T^{2} \)
89 \( 1 - 62.9T + 7.92e3T^{2} \)
97 \( 1 - 91.3iT - 9.40e3T^{2} \)
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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

−9.902899799933627511106434721470, −9.151729793521780117679644551369, −7.893748096939856031476493774883, −7.79168422751777807929111963810, −6.55392665073970668890487906134, −5.75701593554078716419604821449, −4.60559995882678091467326009283, −3.14577976013132568227171150479, −2.06917822841894713049346521146, −0.70766537275434384426676424088, 0.891157833263681942520345168106, 2.16398443661321986896796673085, 3.30058904354117693898419726205, 4.55446154459517092450108748718, 5.92916460279235074391499509844, 6.64334148325965332220179303187, 7.52262271200959473199918607511, 8.550265203253475447840965207121, 8.923613683115220673856866372486, 9.898721017200312584442706718722

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