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Properties

Label 2-30e2-36.31-c0-0-0
Degree $2$
Conductor $900$
Sign $0.939 + 0.342i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s − 0.999·21-s + (0.866 + 0.5i)23-s + (0.499 − 0.866i)24-s − 0.999i·27-s − 0.999i·28-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s − 0.999·21-s + (0.866 + 0.5i)23-s + (0.499 − 0.866i)24-s − 0.999i·27-s − 0.999i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5677749731\)
\(L(\frac12)\) \(\approx\) \(0.5677749731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−10.42700848390621985474927384794, −9.394435600847425396542626112748, −8.449296826747297693673260201261, −7.46699946029881942192492962638, −7.19817905569662309929529293930, −5.99723983351032494088585004851, −5.34042349022179951479056514552, −4.30177137695258627828954065354, −2.22188553346566853230913988191, −1.02067260218162582006938144824, 1.29540585027610531385593859983, 2.75969226503359838135478581940, 4.08113616362854792679244335778, 5.02573130791424161470127936589, 6.07712454519151878211262297271, 7.06771007184409688590845993697, 7.974913179066763630309007449262, 8.961920343272576124571207253254, 9.489595480128461231313880652289, 10.57412507204392932604122735996

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