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Properties

Label 2-30e2-4.3-c2-0-82
Degree $2$
Conductor $900$
Sign $-0.0876 + 0.996i$
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  + (1.35 + 1.47i)2-s + (−0.350 + 3.98i)4-s − 10.9i·7-s + (−6.35 + 4.86i)8-s + 11.3i·11-s − 10.8·13-s + (16.1 − 14.7i)14-s + (−15.7 − 2.79i)16-s − 15.8·17-s − 24.9i·19-s + (−16.7 + 15.3i)22-s − 20.9i·23-s + (−14.5 − 15.9i)26-s + (43.5 + 3.83i)28-s − 22.8·29-s + ⋯
L(s)  = 1  + (0.675 + 0.737i)2-s + (−0.0876 + 0.996i)4-s − 1.55i·7-s + (−0.793 + 0.608i)8-s + 1.03i·11-s − 0.831·13-s + (1.15 − 1.05i)14-s + (−0.984 − 0.174i)16-s − 0.929·17-s − 1.31i·19-s + (−0.761 + 0.697i)22-s − 0.911i·23-s + (−0.561 − 0.613i)26-s + (1.55 + 0.136i)28-s − 0.786·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0876 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0876 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0876 + 0.996i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.0876 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6985127399\)
\(L(\frac12)\) \(\approx\) \(0.6985127399\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 1.47i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 10.9iT - 49T^{2} \)
11 \( 1 - 11.3iT - 121T^{2} \)
13 \( 1 + 10.8T + 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
19 \( 1 + 24.9iT - 361T^{2} \)
23 \( 1 + 20.9iT - 529T^{2} \)
29 \( 1 + 22.8T + 841T^{2} \)
31 \( 1 - 22.7iT - 961T^{2} \)
37 \( 1 + 19.1T + 1.36e3T^{2} \)
41 \( 1 + 17T + 1.68e3T^{2} \)
43 \( 1 + 6.51iT - 1.84e3T^{2} \)
47 \( 1 + 38.9iT - 2.20e3T^{2} \)
53 \( 1 + 13.2T + 2.80e3T^{2} \)
59 \( 1 + 95.6iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + 54.1iT - 4.48e3T^{2} \)
71 \( 1 - 68.5iT - 5.04e3T^{2} \)
73 \( 1 - 44.1T + 5.32e3T^{2} \)
79 \( 1 - 81.7iT - 6.24e3T^{2} \)
83 \( 1 + 27.9iT - 6.88e3T^{2} \)
89 \( 1 - 42.1T + 7.92e3T^{2} \)
97 \( 1 - 154.T + 9.40e3T^{2} \)
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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

−9.637779290897989907423547841399, −8.711966776705164994658765473004, −7.61029958217588109238355087278, −7.03433670030860530347853457160, −6.55709512869970238916668453327, −4.92862177822437517757439413291, −4.59259430510740993080177222119, −3.55507707422690274208430481141, −2.22227904354351068322235291464, −0.16459720036137021684057819183, 1.73861561880539784939769040706, 2.67906678990703307554251980351, 3.63600184556765564603502359295, 4.86533644235862988177321484337, 5.75638171654694553603062133063, 6.18417806231336820251277395199, 7.64310836701570980159401025520, 8.798195697214764417183563472445, 9.303306162578511376322075881380, 10.23190233503081071237903361108

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