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Properties

Label 2-30e2-20.19-c2-0-58
Degree $2$
Conductor $900$
Sign $0.722 + 0.691i$
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  + (−1.97 + 0.342i)2-s + (3.76 − 1.34i)4-s + 11.5·7-s + (−6.95 + 3.94i)8-s + 9.97i·11-s − 14.1i·13-s + (−22.6 + 3.94i)14-s + (12.3 − 10.1i)16-s − 30.5i·17-s − 12.2i·19-s + (−3.41 − 19.6i)22-s + 15.7·23-s + (4.83 + 27.8i)26-s + (43.3 − 15.5i)28-s + 18.8·29-s + ⋯
L(s)  = 1  + (−0.985 + 0.171i)2-s + (0.941 − 0.337i)4-s + 1.64·7-s + (−0.869 + 0.493i)8-s + 0.906i·11-s − 1.08i·13-s + (−1.62 + 0.281i)14-s + (0.772 − 0.635i)16-s − 1.79i·17-s − 0.643i·19-s + (−0.155 − 0.893i)22-s + 0.685·23-s + (0.185 + 1.07i)26-s + (1.54 − 0.554i)28-s + 0.651·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.722 + 0.691i$
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: \((2,\ 900,\ (\ :1),\ 0.722 + 0.691i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.409135738\)
\(L(\frac12)\) \(\approx\) \(1.409135738\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.97 - 0.342i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 11.5T + 49T^{2} \)
11 \( 1 - 9.97iT - 121T^{2} \)
13 \( 1 + 14.1iT - 169T^{2} \)
17 \( 1 + 30.5iT - 289T^{2} \)
19 \( 1 + 12.2iT - 361T^{2} \)
23 \( 1 - 15.7T + 529T^{2} \)
29 \( 1 - 18.8T + 841T^{2} \)
31 \( 1 - 35.2iT - 961T^{2} \)
37 \( 1 + 50.1iT - 1.36e3T^{2} \)
41 \( 1 + 28.8T + 1.68e3T^{2} \)
43 \( 1 + 24.4T + 1.84e3T^{2} \)
47 \( 1 + 55.6T + 2.20e3T^{2} \)
53 \( 1 + 2.46iT - 2.80e3T^{2} \)
59 \( 1 + 64.6iT - 3.48e3T^{2} \)
61 \( 1 + 30.2T + 3.72e3T^{2} \)
67 \( 1 - 66.1T + 4.48e3T^{2} \)
71 \( 1 + 11.5iT - 5.04e3T^{2} \)
73 \( 1 - 2.24iT - 5.32e3T^{2} \)
79 \( 1 - 78.4iT - 6.24e3T^{2} \)
83 \( 1 - 146.T + 6.88e3T^{2} \)
89 \( 1 + 87.4T + 7.92e3T^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−9.750705912398848713121610936501, −8.905047997985353191068583735970, −8.121228679772737359485835433697, −7.42396584053683680602916531474, −6.77238672036452374915705696789, −5.19408548348090075096460707236, −4.92288756257870933556016815726, −2.99274120651920058517488614835, −1.90930137425510696094058703314, −0.70690757267547242432692705181, 1.25426044076713656301163480876, 2.00328408590846209386263181513, 3.51208995791061708836045774048, 4.62551455487023727121937518962, 5.89570158124271640206572178302, 6.69387729737629634375202831510, 7.929883865464612018539698962406, 8.288759801757138666374723019343, 8.970207927848407195522144247283, 10.11773785375908093969801479777

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