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Properties

Label 2-30e2-9.5-c2-0-30
Degree $2$
Conductor $900$
Sign $-0.640 + 0.768i$
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  + (−1.92 + 2.30i)3-s + (0.382 + 0.662i)7-s + (−1.59 − 8.85i)9-s + (6.98 − 4.03i)11-s + (1.08 − 1.87i)13-s + 24.9i·17-s − 20.7·19-s + (−2.25 − 0.394i)21-s + (−14.5 − 8.42i)23-s + (23.4 + 13.3i)27-s + (−10.1 + 5.86i)29-s + (−0.334 + 0.579i)31-s + (−4.16 + 23.8i)33-s − 48.2·37-s + (2.23 + 6.10i)39-s + ⋯
L(s)  = 1  + (−0.641 + 0.767i)3-s + (0.0546 + 0.0946i)7-s + (−0.176 − 0.984i)9-s + (0.635 − 0.366i)11-s + (0.0833 − 0.144i)13-s + 1.46i·17-s − 1.09·19-s + (−0.107 − 0.0187i)21-s + (−0.634 − 0.366i)23-s + (0.868 + 0.495i)27-s + (−0.350 + 0.202i)29-s + (−0.0107 + 0.0187i)31-s + (−0.126 + 0.722i)33-s − 1.30·37-s + (0.0572 + 0.156i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.640 + 0.768i$
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: \((2,\ 900,\ (\ :1),\ -0.640 + 0.768i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1367580096\)
\(L(\frac12)\) \(\approx\) \(0.1367580096\)
\(L(2)\) 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 \( 1 + (1.92 - 2.30i)T \)
5 \( 1 \)
good7 \( 1 + (-0.382 - 0.662i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.98 + 4.03i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.08 + 1.87i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 24.9iT - 289T^{2} \)
19 \( 1 + 20.7T + 361T^{2} \)
23 \( 1 + (14.5 + 8.42i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (10.1 - 5.86i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (0.334 - 0.579i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 48.2T + 1.36e3T^{2} \)
41 \( 1 + (54.1 + 31.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-35.6 - 61.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (17.5 - 10.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 82.8iT - 2.80e3T^{2} \)
59 \( 1 + (4.66 + 2.69i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41.6 + 72.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5.31 - 9.20i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 87.9iT - 5.04e3T^{2} \)
73 \( 1 + 15.5T + 5.32e3T^{2} \)
79 \( 1 + (-16.8 - 29.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-48.2 + 27.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 22.3iT - 7.92e3T^{2} \)
97 \( 1 + (34.4 + 59.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.736480343561258213853011539905, −8.781760556924087863743855343363, −8.192802847353663225908733346985, −6.68939872324357397387319101743, −6.15710635789413745376329597503, −5.20899544350826667053805741826, −4.14689393639595723100052166692, −3.46315022758795891698691930671, −1.74633194959848225943433260891, −0.04886587362275437523705817020, 1.39781229053347000021163207786, 2.52039271937428299577940677527, 4.05081501404203171170362244776, 5.04486419450757243007274345253, 5.98127609051494631234706964883, 6.88019599794417602445839620278, 7.42803057630526213022534307323, 8.483780786086556233854877316702, 9.350779669377534528054068772893, 10.36702257836941015090522395971

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