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Properties

Label 2-30e2-45.29-c2-0-26
Degree $2$
Conductor $900$
Sign $0.856 + 0.515i$
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  + (2.96 + 0.459i)3-s + (−7.16 − 4.13i)7-s + (8.57 + 2.72i)9-s + (3.29 + 1.90i)11-s + (16.7 − 9.65i)13-s − 7.87·17-s + 12.3·19-s + (−19.3 − 15.5i)21-s + (0.604 + 1.04i)23-s + (24.1 + 12.0i)27-s + (−10.7 − 6.20i)29-s + (−4.65 − 8.06i)31-s + (8.89 + 7.15i)33-s − 37.0i·37-s + (54.0 − 20.9i)39-s + ⋯
L(s)  = 1  + (0.988 + 0.153i)3-s + (−1.02 − 0.590i)7-s + (0.953 + 0.302i)9-s + (0.299 + 0.172i)11-s + (1.28 − 0.742i)13-s − 0.463·17-s + 0.649·19-s + (−0.920 − 0.740i)21-s + (0.0262 + 0.0455i)23-s + (0.895 + 0.444i)27-s + (−0.370 − 0.213i)29-s + (−0.150 − 0.260i)31-s + (0.269 + 0.216i)33-s − 1.00i·37-s + (1.38 − 0.537i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.856 + 0.515i$
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: \((2,\ 900,\ (\ :1),\ 0.856 + 0.515i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.626475562\)
\(L(\frac12)\) \(\approx\) \(2.626475562\)
\(L(2)\) 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 \( 1 + (-2.96 - 0.459i)T \)
5 \( 1 \)
good7 \( 1 + (7.16 + 4.13i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.29 - 1.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-16.7 + 9.65i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 7.87T + 289T^{2} \)
19 \( 1 - 12.3T + 361T^{2} \)
23 \( 1 + (-0.604 - 1.04i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (10.7 + 6.20i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (4.65 + 8.06i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 37.0iT - 1.36e3T^{2} \)
41 \( 1 + (-40.1 + 23.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-53.1 - 30.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-46.0 + 79.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 52.1T + 2.80e3T^{2} \)
59 \( 1 + (-54.4 + 31.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11.6 + 20.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-93.8 + 54.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 134. iT - 5.04e3T^{2} \)
73 \( 1 + 66.7iT - 5.32e3T^{2} \)
79 \( 1 + (62.2 - 107. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-40.3 + 69.8i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 176. iT - 7.92e3T^{2} \)
97 \( 1 + (79.5 + 45.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.676787517025694074853406361303, −9.119147975656602675598126298820, −8.203399520652517179178213542864, −7.35973546635258153220641197979, −6.56577344662806351154425646157, −5.50332061424897948738832559455, −3.99427508376317722770056802946, −3.58590703766679414931426603457, −2.40801712223066788600643562697, −0.875855096100948615179718744846, 1.26419253168295413145316836892, 2.59517056482454593058742723521, 3.47144786247122204786672586751, 4.34608931110439128510557728466, 5.89345265941634239568961514671, 6.56146613899403979753121391446, 7.46791605855256765838578323095, 8.536447052577838529125033413094, 9.138252194003456015649267219700, 9.621774699571024763919322181340

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