Properties

Label 2-30e2-9.5-c2-0-24
Degree $2$
Conductor $900$
Sign $0.969 + 0.246i$
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  + (2.85 − 0.920i)3-s + (−0.594 − 1.02i)7-s + (7.30 − 5.25i)9-s + (3.63 − 2.09i)11-s + (−7.15 + 12.3i)13-s + 18.7i·17-s + 33.8·19-s + (−2.64 − 2.39i)21-s + (11.6 + 6.74i)23-s + (16.0 − 21.7i)27-s + (30.6 − 17.6i)29-s + (4.97 − 8.62i)31-s + (8.44 − 9.33i)33-s + 19.3·37-s + (−9.02 + 41.9i)39-s + ⋯
L(s)  = 1  + (0.951 − 0.306i)3-s + (−0.0849 − 0.147i)7-s + (0.811 − 0.584i)9-s + (0.330 − 0.190i)11-s + (−0.550 + 0.953i)13-s + 1.10i·17-s + 1.78·19-s + (−0.125 − 0.113i)21-s + (0.508 + 0.293i)23-s + (0.593 − 0.805i)27-s + (1.05 − 0.609i)29-s + (0.160 − 0.278i)31-s + (0.255 − 0.282i)33-s + 0.521·37-s + (−0.231 + 1.07i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.878302148\)
\(L(\frac12)\) \(\approx\) \(2.878302148\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.85 + 0.920i)T \)
5 \( 1 \)
good7 \( 1 + (0.594 + 1.02i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.63 + 2.09i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.15 - 12.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 18.7iT - 289T^{2} \)
19 \( 1 - 33.8T + 361T^{2} \)
23 \( 1 + (-11.6 - 6.74i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-30.6 + 17.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-4.97 + 8.62i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 19.3T + 1.36e3T^{2} \)
41 \( 1 + (55.9 + 32.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (20.7 + 35.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-58.2 + 33.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 30.0iT - 2.80e3T^{2} \)
59 \( 1 + (-3.66 - 2.11i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-43.8 - 75.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (17.9 - 31.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 24.5iT - 5.04e3T^{2} \)
73 \( 1 + 11.9T + 5.32e3T^{2} \)
79 \( 1 + (-19.0 - 33.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (20.8 - 12.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 44.4iT - 7.92e3T^{2} \)
97 \( 1 + (-31.9 - 55.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.757548587439649465065106492962, −8.995671482764581155684010288836, −8.277466615888309603724877411725, −7.29074783847524542802336044852, −6.75164718217858146571142570622, −5.53184287941545043324383163437, −4.26401356152411208763868510790, −3.44718736243029650077319285089, −2.30036573524464547069038900167, −1.11517477485590296002928295154, 1.11265401217483346983450565177, 2.73641931983528551985856186637, 3.24580415646980677519673895731, 4.66219005321388122572685381385, 5.29705151755803214109905743598, 6.76061308292081529404362388355, 7.52945359160108280544577643187, 8.255504967023463378308350253791, 9.273612396533837125849284472325, 9.737510258062827066100983743860

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