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Properties

Label 2-30e2-45.29-c2-0-10
Degree $2$
Conductor $900$
Sign $-0.613 - 0.789i$
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  + (2.37 + 1.83i)3-s + (7.15 + 4.13i)7-s + (2.23 + 8.71i)9-s + (−1.19 − 0.687i)11-s + (−18.6 + 10.7i)13-s − 25.3·17-s + 2.49·19-s + (9.36 + 22.9i)21-s + (19.1 + 33.1i)23-s + (−10.7 + 24.7i)27-s + (−37.0 − 21.4i)29-s + (10.9 + 18.9i)31-s + (−1.55 − 3.81i)33-s − 30.5i·37-s + (−64.0 − 8.78i)39-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)3-s + (1.02 + 0.590i)7-s + (0.248 + 0.968i)9-s + (−0.108 − 0.0624i)11-s + (−1.43 + 0.829i)13-s − 1.48·17-s + 0.131·19-s + (0.445 + 1.09i)21-s + (0.833 + 1.44i)23-s + (−0.397 + 0.917i)27-s + (−1.27 − 0.738i)29-s + (0.353 + 0.611i)31-s + (−0.0472 − 0.115i)33-s − 0.826i·37-s + (−1.64 − 0.225i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.613 - 0.789i$
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: \((2,\ 900,\ (\ :1),\ -0.613 - 0.789i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.083057977\)
\(L(\frac12)\) \(\approx\) \(2.083057977\)
\(L(2)\) 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 \( 1 + (-2.37 - 1.83i)T \)
5 \( 1 \)
good7 \( 1 + (-7.15 - 4.13i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.19 + 0.687i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (18.6 - 10.7i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 25.3T + 289T^{2} \)
19 \( 1 - 2.49T + 361T^{2} \)
23 \( 1 + (-19.1 - 33.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (37.0 + 21.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-10.9 - 18.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 30.5iT - 1.36e3T^{2} \)
41 \( 1 + (-7.29 + 4.21i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-61.6 - 35.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-24.3 + 42.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 1.96T + 2.80e3T^{2} \)
59 \( 1 + (3.77 - 2.18i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (18.4 - 32.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.9 - 8.06i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 71.5iT - 5.04e3T^{2} \)
73 \( 1 - 122. iT - 5.32e3T^{2} \)
79 \( 1 + (3.98 - 6.90i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-52.1 + 90.2i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 9.37iT - 7.92e3T^{2} \)
97 \( 1 + (-11.8 - 6.86i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.02377339210254582319804541019, −9.183895767011742823382655888814, −8.808907680528923468190279660672, −7.67801275625802422305509451227, −7.14011131398265549585996966368, −5.57320147729239636405475855234, −4.79815583030862780793674818219, −4.03982528134842245892328536858, −2.59594398624591959584944825363, −1.89714110066784997560732256261, 0.57123692541851096928503499176, 2.00138703567392008653973444391, 2.85452731393091043098160083172, 4.26908953146848858558220204955, 4.99506630450933089083338069311, 6.39169663857694501022046514530, 7.36337917515887959519279586360, 7.75086495108208754201960549313, 8.721268711839506106543785280597, 9.420643174450855770476564080192

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