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Properties

Label 2-30e2-4.3-c2-0-45
Degree $2$
Conductor $900$
Sign $0.548 + 0.836i$
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  + (0.950 − 1.75i)2-s + (−2.19 − 3.34i)4-s + 3.98i·7-s + (−7.97 + 0.677i)8-s + 8.39i·11-s + 9.77·13-s + (7.02 + 3.79i)14-s + (−6.38 + 14.6i)16-s + 12.1·17-s − 17.3i·19-s + (14.7 + 7.97i)22-s + 2.97i·23-s + (9.28 − 17.1i)26-s + (13.3 − 8.74i)28-s + 51.6·29-s + ⋯
L(s)  = 1  + (0.475 − 0.879i)2-s + (−0.548 − 0.836i)4-s + 0.569i·7-s + (−0.996 + 0.0847i)8-s + 0.763i·11-s + 0.751·13-s + (0.501 + 0.270i)14-s + (−0.399 + 0.916i)16-s + 0.714·17-s − 0.914i·19-s + (0.671 + 0.362i)22-s + 0.129i·23-s + (0.357 − 0.661i)26-s + (0.476 − 0.312i)28-s + 1.78·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.548 + 0.836i$
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: \((2,\ 900,\ (\ :1),\ 0.548 + 0.836i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.322738529\)
\(L(\frac12)\) \(\approx\) \(2.322738529\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.950 + 1.75i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 3.98iT - 49T^{2} \)
11 \( 1 - 8.39iT - 121T^{2} \)
13 \( 1 - 9.77T + 169T^{2} \)
17 \( 1 - 12.1T + 289T^{2} \)
19 \( 1 + 17.3iT - 361T^{2} \)
23 \( 1 - 2.97iT - 529T^{2} \)
29 \( 1 - 51.6T + 841T^{2} \)
31 \( 1 - 30.7iT - 961T^{2} \)
37 \( 1 - 19.5T + 1.36e3T^{2} \)
41 \( 1 - 42.5T + 1.68e3T^{2} \)
43 \( 1 + 62.9iT - 1.84e3T^{2} \)
47 \( 1 + 39.5iT - 2.20e3T^{2} \)
53 \( 1 + 21.2T + 2.80e3T^{2} \)
59 \( 1 - 50.3iT - 3.48e3T^{2} \)
61 \( 1 + 70.3T + 3.72e3T^{2} \)
67 \( 1 + 94.8iT - 4.48e3T^{2} \)
71 \( 1 - 132. iT - 5.04e3T^{2} \)
73 \( 1 - 77.7T + 5.32e3T^{2} \)
79 \( 1 + 48.3iT - 6.24e3T^{2} \)
83 \( 1 - 140. iT - 6.88e3T^{2} \)
89 \( 1 + 18.1T + 7.92e3T^{2} \)
97 \( 1 + 15T + 9.40e3T^{2} \)
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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

−9.956105264870542662661128277892, −9.068542394905121031741377225499, −8.417905226154049172303538482123, −7.08495349622813148394859433806, −6.09980958796514854210508412518, −5.20307447523257373109463799279, −4.36108589131096922397632623144, −3.22799045532659065580972226982, −2.27048377859529575682372402022, −0.979098558218005957658703365437, 0.902569500989783783505077437076, 2.95206069112715233914091515470, 3.86230222008088436260276432673, 4.74289587101808273766197926481, 5.99456009344897243705278694057, 6.31118886506117403500732487069, 7.64563009639044758402773485845, 8.057815018351082934657560230365, 9.002386940637377379810226916111, 9.945202445683354929879785292864

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