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Properties

Label 2-30e2-45.29-c2-0-29
Degree $2$
Conductor $900$
Sign $-0.213 + 0.977i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.213 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.213 + 0.977i$
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: \((2,\ 900,\ (\ :1),\ -0.213 + 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4142143730\)
\(L(\frac12)\) \(\approx\) \(0.4142143730\)
\(L(2)\) 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

−9.774457241329791409310185579652, −8.731322580657679140734901156692, −8.417243505212178485820116869914, −6.90027128645275639792333791664, −6.13128843142135501114307788912, −5.21010870788814065428068917810, −4.23100037906543537509768817887, −3.48843247080687168165560569887, −2.01576234113173148885324093939, −0.13811343064582982781536719100, 1.39832801480390835280684054063, 2.39472247818085561118959043622, 3.86863607877400271588736982272, 4.89527315324997117596559810422, 6.21759745091058368204640750777, 6.52568485678907446000978405741, 7.52179145399815879561256102939, 8.672017372822207254217996776406, 8.847984685810121047887233398833, 10.42778488615810715324705653030

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