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Properties

Label 2-30e2-45.14-c2-0-29
Degree $2$
Conductor $900$
Sign $-0.239 + 0.970i$
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  + (0.708 − 2.91i)3-s + (8.59 − 4.96i)7-s + (−7.99 − 4.12i)9-s + (3.59 − 2.07i)11-s + (1.06 + 0.613i)13-s + 21.0·17-s + 9.84·19-s + (−8.38 − 28.5i)21-s + (1.34 − 2.32i)23-s + (−17.6 + 20.3i)27-s + (0.321 − 0.185i)29-s + (22.1 − 38.4i)31-s + (−3.50 − 11.9i)33-s − 33.0i·37-s + (2.53 − 2.66i)39-s + ⋯
L(s)  = 1  + (0.236 − 0.971i)3-s + (1.22 − 0.708i)7-s + (−0.888 − 0.458i)9-s + (0.326 − 0.188i)11-s + (0.0817 + 0.0471i)13-s + 1.23·17-s + 0.517·19-s + (−0.399 − 1.36i)21-s + (0.0582 − 0.100i)23-s + (−0.655 + 0.755i)27-s + (0.0110 − 0.00640i)29-s + (0.715 − 1.23i)31-s + (−0.106 − 0.361i)33-s − 0.893i·37-s + (0.0651 − 0.0682i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.239 + 0.970i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.239 + 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.383723571\)
\(L(\frac12)\) \(\approx\) \(2.383723571\)
\(L(2)\) 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 \( 1 + (-0.708 + 2.91i)T \)
5 \( 1 \)
good7 \( 1 + (-8.59 + 4.96i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.59 + 2.07i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.06 - 0.613i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 21.0T + 289T^{2} \)
19 \( 1 - 9.84T + 361T^{2} \)
23 \( 1 + (-1.34 + 2.32i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.321 + 0.185i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-22.1 + 38.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 33.0iT - 1.36e3T^{2} \)
41 \( 1 + (34.2 + 19.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (25.2 - 14.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.6 - 68.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 92.8T + 2.80e3T^{2} \)
59 \( 1 + (-53.3 - 30.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-12.6 - 21.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-31.0 - 17.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.7iT - 5.04e3T^{2} \)
73 \( 1 + 62.4iT - 5.32e3T^{2} \)
79 \( 1 + (71.9 + 124. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-8.20 - 14.2i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 155. iT - 7.92e3T^{2} \)
97 \( 1 + (-90.6 + 52.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.598795259463665347398623316553, −8.603526088057559212463582128095, −7.78300633795232365444421321980, −7.42355059899415467884168686265, −6.27941058578211813100981167422, −5.38837877245469886013777250644, −4.23505999224364780090687287438, −3.07953288648791307868425900195, −1.72094611036504493968033306391, −0.834424987702338594796828687394, 1.48629763659144654288748339019, 2.83522115201247335888042833905, 3.84399543776657682838356178799, 5.09629941835669640892989149441, 5.30952546053551536090039728763, 6.69899657716843436928045109450, 8.064993950110068195761461417121, 8.365276962403714815489838973970, 9.391841339649562395239055101870, 10.07671416262366691293062434567

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