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Properties

Label 2-30e2-45.29-c2-0-22
Degree $2$
Conductor $900$
Sign $0.416 + 0.909i$
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  + (−2.83 + 0.978i)3-s + (4.69 + 2.71i)7-s + (7.08 − 5.54i)9-s + (−8.81 − 5.09i)11-s + (4.42 − 2.55i)13-s − 17.4·17-s + 17.4·19-s + (−15.9 − 3.09i)21-s + (9.41 + 16.3i)23-s + (−14.6 + 22.6i)27-s + (−29.0 − 16.7i)29-s + (−25.4 − 44.1i)31-s + (29.9 + 5.81i)33-s − 0.605i·37-s + (−10.0 + 11.5i)39-s + ⋯
L(s)  = 1  + (−0.945 + 0.326i)3-s + (0.671 + 0.387i)7-s + (0.787 − 0.616i)9-s + (−0.801 − 0.462i)11-s + (0.340 − 0.196i)13-s − 1.02·17-s + 0.920·19-s + (−0.760 − 0.147i)21-s + (0.409 + 0.709i)23-s + (−0.543 + 0.839i)27-s + (−1.00 − 0.578i)29-s + (−0.822 − 1.42i)31-s + (0.908 + 0.176i)33-s − 0.0163i·37-s + (−0.257 + 0.296i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.416 + 0.909i$
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: \((2,\ 900,\ (\ :1),\ 0.416 + 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9999242025\)
\(L(\frac12)\) \(\approx\) \(0.9999242025\)
\(L(2)\) 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 \( 1 + (2.83 - 0.978i)T \)
5 \( 1 \)
good7 \( 1 + (-4.69 - 2.71i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.81 + 5.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.42 + 2.55i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 17.4T + 289T^{2} \)
19 \( 1 - 17.4T + 361T^{2} \)
23 \( 1 + (-9.41 - 16.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (29.0 + 16.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (25.4 + 44.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 0.605iT - 1.36e3T^{2} \)
41 \( 1 + (12.4 - 7.17i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-44.0 - 25.4i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.13 - 1.96i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 18.4T + 2.80e3T^{2} \)
59 \( 1 + (-70.8 + 40.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-4.70 + 8.14i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-31.8 + 18.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 57.2iT - 5.04e3T^{2} \)
73 \( 1 + 78.9iT - 5.32e3T^{2} \)
79 \( 1 + (-19.1 + 33.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-38.6 + 66.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 138. iT - 7.92e3T^{2} \)
97 \( 1 + (101. + 58.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.781624263940454113131443044138, −9.083870364102018377734516461547, −7.988116259921828809019144973703, −7.22592542633089053596132075231, −6.00093414862671658664062083457, −5.45606870886931763082307816976, −4.59013262915210176457063501350, −3.46305424639011278626723704278, −1.96260548126661093097166207436, −0.42459812086619264104170905878, 1.10588676795560288333538262549, 2.29477121945595149521245434456, 3.95844517501598709422920905416, 4.97171618086732584323265497533, 5.52754389902536672979107670337, 6.84247189683856057814196822483, 7.26966481172083117119399055939, 8.267461927688969013473805346840, 9.260303807599972632979911577206, 10.40320758768238648415966127809

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