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Properties

Label 2-30e2-20.19-c2-0-81
Degree $2$
Conductor $900$
Sign $0.997 + 0.0769i$
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  + (1.92 + 0.534i)2-s + (3.42 + 2.05i)4-s + 11.9·7-s + (5.51 + 5.79i)8-s − 14.5i·11-s − 22.4i·13-s + (23.0 + 6.39i)14-s + (7.52 + 14.1i)16-s − 12.6i·17-s + 8.76i·19-s + (7.76 − 28.0i)22-s − 4.99·23-s + (12.0 − 43.3i)26-s + (41.0 + 24.6i)28-s + 2.74·29-s + ⋯
L(s)  = 1  + (0.963 + 0.267i)2-s + (0.857 + 0.514i)4-s + 1.71·7-s + (0.688 + 0.724i)8-s − 1.32i·11-s − 1.72i·13-s + (1.64 + 0.456i)14-s + (0.470 + 0.882i)16-s − 0.746i·17-s + 0.461i·19-s + (0.352 − 1.27i)22-s − 0.217·23-s + (0.461 − 1.66i)26-s + (1.46 + 0.880i)28-s + 0.0947·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.997 + 0.0769i$
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: \((2,\ 900,\ (\ :1),\ 0.997 + 0.0769i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.363088629\)
\(L(\frac12)\) \(\approx\) \(4.363088629\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.92 - 0.534i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 11.9T + 49T^{2} \)
11 \( 1 + 14.5iT - 121T^{2} \)
13 \( 1 + 22.4iT - 169T^{2} \)
17 \( 1 + 12.6iT - 289T^{2} \)
19 \( 1 - 8.76iT - 361T^{2} \)
23 \( 1 + 4.99T + 529T^{2} \)
29 \( 1 - 2.74T + 841T^{2} \)
31 \( 1 + 16.3iT - 961T^{2} \)
37 \( 1 - 32.4iT - 1.36e3T^{2} \)
41 \( 1 + 42.7T + 1.68e3T^{2} \)
43 \( 1 - 16.5T + 1.84e3T^{2} \)
47 \( 1 + 48.5T + 2.20e3T^{2} \)
53 \( 1 - 94.1iT - 2.80e3T^{2} \)
59 \( 1 - 43.2iT - 3.48e3T^{2} \)
61 \( 1 - 56.7T + 3.72e3T^{2} \)
67 \( 1 + 61.1T + 4.48e3T^{2} \)
71 \( 1 + 39.6iT - 5.04e3T^{2} \)
73 \( 1 - 99.5iT - 5.32e3T^{2} \)
79 \( 1 - 10.7iT - 6.24e3T^{2} \)
83 \( 1 - 140.T + 6.88e3T^{2} \)
89 \( 1 - 54.8T + 7.92e3T^{2} \)
97 \( 1 - 14.1iT - 9.40e3T^{2} \)
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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

−10.26550469202487742670697080382, −8.637495258233760140663939214468, −8.038529357025602014902459261716, −7.50976185383828411525270353656, −6.14024056415895074696941447406, −5.41469427415555535832901268727, −4.77727596159078536491205548857, −3.57085877332959698049165680733, −2.60205698398843796210395251461, −1.14438824681343145317772979501, 1.68433979146283712418535939617, 2.06163413897362031791847746559, 3.85362430817767310107702921747, 4.64361247110562166744784414871, 5.12929108059076830347103269589, 6.48983970140140157392023065811, 7.17753024403504285799047173973, 8.092957777467003157600348806741, 9.162808002757505076232470887789, 10.15965719653552137081661394345

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