Properties

Label 2-30e2-4.3-c2-0-53
Degree $2$
Conductor $900$
Sign $0.875 - 0.484i$
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  + (0.5 + 1.93i)2-s + (−3.50 + 1.93i)4-s + (−5.50 − 5.80i)8-s + (8.50 − 13.5i)16-s + 14·17-s − 30.9i·19-s − 30.9i·23-s + 61.9i·31-s + (30.5 + 9.68i)32-s + (7 + 27.1i)34-s + (60.0 − 15.4i)38-s + (60.0 − 15.4i)46-s − 92.9i·47-s + 49·49-s + 86·53-s + ⋯
L(s)  = 1  + (0.250 + 0.968i)2-s + (−0.875 + 0.484i)4-s + (−0.687 − 0.726i)8-s + (0.531 − 0.847i)16-s + 0.823·17-s − 1.63i·19-s − 1.34i·23-s + 1.99i·31-s + (0.953 + 0.302i)32-s + (0.205 + 0.797i)34-s + (1.57 − 0.407i)38-s + (1.30 − 0.336i)46-s − 1.97i·47-s + 0.999·49-s + 1.62·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.875 - 0.484i$
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: \((2,\ 900,\ (\ :1),\ 0.875 - 0.484i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.748149278\)
\(L(\frac12)\) \(\approx\) \(1.748149278\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 1.93i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 14T + 289T^{2} \)
19 \( 1 + 30.9iT - 361T^{2} \)
23 \( 1 + 30.9iT - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 - 61.9iT - 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + 92.9iT - 2.20e3T^{2} \)
53 \( 1 - 86T + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 118T + 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 + 61.9iT - 6.88e3T^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 + 9.40e3T^{2} \)
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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

−9.864473179883106180022424344895, −8.769321032484791338039519622630, −8.426564959904299688452739566144, −7.10490609286310707001461012683, −6.80250495111581684565540076251, −5.55168824246522250293868340000, −4.88701795314231003894735663404, −3.82749204806894224196570705590, −2.68617202105089289588207724228, −0.65790968067746589783569440519, 1.05562944383288945052537382242, 2.22972943008531047637106077038, 3.47715187727858018604461441177, 4.17319792296347694424780866894, 5.48103353130138898002890199466, 5.97865886918113664478881843963, 7.52689278994781506594991910252, 8.227685420935828999900138046427, 9.364562297957143705828456821191, 9.891384690975723758663312508526

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