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Properties

Label 2-30e2-45.29-c2-0-35
Degree $2$
Conductor $900$
Sign $-0.653 - 0.756i$
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  + (0.920 − 2.85i)3-s + (−1.02 − 0.594i)7-s + (−7.30 − 5.25i)9-s + (3.63 + 2.09i)11-s + (−12.3 + 7.15i)13-s + 18.7·17-s − 33.8·19-s + (−2.64 + 2.39i)21-s + (−6.74 − 11.6i)23-s + (−21.7 + 16.0i)27-s + (−30.6 − 17.6i)29-s + (4.97 + 8.62i)31-s + (9.33 − 8.44i)33-s + 19.3i·37-s + (9.02 + 41.9i)39-s + ⋯
L(s)  = 1  + (0.306 − 0.951i)3-s + (−0.147 − 0.0849i)7-s + (−0.811 − 0.584i)9-s + (0.330 + 0.190i)11-s + (−0.953 + 0.550i)13-s + 1.10·17-s − 1.78·19-s + (−0.125 + 0.113i)21-s + (−0.293 − 0.508i)23-s + (−0.805 + 0.593i)27-s + (−1.05 − 0.609i)29-s + (0.160 + 0.278i)31-s + (0.282 − 0.255i)33-s + 0.521i·37-s + (0.231 + 1.07i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.653 - 0.756i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.653 - 0.756i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01919419241\)
\(L(\frac12)\) \(\approx\) \(0.01919419241\)
\(L(2)\) 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 \( 1 + (-0.920 + 2.85i)T \)
5 \( 1 \)
good7 \( 1 + (1.02 + 0.594i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.63 - 2.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (12.3 - 7.15i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 18.7T + 289T^{2} \)
19 \( 1 + 33.8T + 361T^{2} \)
23 \( 1 + (6.74 + 11.6i)T + (-264.5 + 458. i)T^{2} \)
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 \( 1 - 19.3iT - 1.36e3T^{2} \)
41 \( 1 + (55.9 - 32.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-35.9 - 20.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (33.6 - 58.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 30.0T + 2.80e3T^{2} \)
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 \( 1 + (-31.0 + 17.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 24.5iT - 5.04e3T^{2} \)
73 \( 1 - 11.9iT - 5.32e3T^{2} \)
79 \( 1 + (19.0 - 33.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (12.0 - 20.8i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 44.4iT - 7.92e3T^{2} \)
97 \( 1 + (-55.3 - 31.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.373252482087208577164885821118, −8.349400062912888819014200440725, −7.74742411954658565416444581432, −6.73536649096710541344560236930, −6.22961338134130526740470338230, −4.94822429124883632722407936772, −3.79511471076157337683613331641, −2.57712034512930770004567075438, −1.59422194787277832209752742486, −0.00554999084463122592705483520, 2.08773874114846425164715043799, 3.26744599884535795108718781490, 4.09533931900623312974263025820, 5.18064654668425140300341131444, 5.89142944701460787015876223760, 7.15296548943413155021010228473, 8.098180870429237525143023501289, 8.848783192411249812539339022757, 9.697870130985979834822412028006, 10.31292822632657963915819273975

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