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Properties

Label 2-30e2-45.29-c2-0-34
Degree $2$
Conductor $900$
Sign $-0.902 + 0.431i$
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  + (2.70 + 1.29i)3-s + (−3.00 − 1.73i)7-s + (5.66 + 6.99i)9-s + (−12.3 − 7.14i)11-s + (−5.28 + 3.04i)13-s − 11.7·17-s − 23.4·19-s + (−5.88 − 8.56i)21-s + (−14.3 − 24.8i)23-s + (6.30 + 26.2i)27-s + (13.3 + 7.68i)29-s + (−18.4 − 32.0i)31-s + (−24.2 − 35.3i)33-s + 46.7i·37-s + (−18.2 + 1.43i)39-s + ⋯
L(s)  = 1  + (0.902 + 0.430i)3-s + (−0.428 − 0.247i)7-s + (0.629 + 0.777i)9-s + (−1.12 − 0.649i)11-s + (−0.406 + 0.234i)13-s − 0.690·17-s − 1.23·19-s + (−0.280 − 0.407i)21-s + (−0.624 − 1.08i)23-s + (0.233 + 0.972i)27-s + (0.459 + 0.265i)29-s + (−0.596 − 1.03i)31-s + (−0.735 − 1.07i)33-s + 1.26i·37-s + (−0.467 + 0.0368i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.902 + 0.431i$
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: \((2,\ 900,\ (\ :1),\ -0.902 + 0.431i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1744243583\)
\(L(\frac12)\) \(\approx\) \(0.1744243583\)
\(L(2)\) 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 \( 1 + (-2.70 - 1.29i)T \)
5 \( 1 \)
good7 \( 1 + (3.00 + 1.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.3 + 7.14i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.28 - 3.04i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 11.7T + 289T^{2} \)
19 \( 1 + 23.4T + 361T^{2} \)
23 \( 1 + (14.3 + 24.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.3 - 7.68i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (18.4 + 32.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 46.7iT - 1.36e3T^{2} \)
41 \( 1 + (17.1 - 9.89i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (4.76 + 2.75i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-5.15 + 8.93i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 41.5T + 2.80e3T^{2} \)
59 \( 1 + (58.5 - 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (26.5 - 45.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.5 + 15.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 2.28iT - 5.04e3T^{2} \)
73 \( 1 + 86.2iT - 5.32e3T^{2} \)
79 \( 1 + (58.4 - 101. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-65.2 + 113. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 75.3iT - 7.92e3T^{2} \)
97 \( 1 + (-24.4 - 14.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.567550582725443871791241883888, −8.587730253648457257044478741214, −8.138549529492210251389250322814, −7.08714584164654875143451539141, −6.15117781552810210944375947353, −4.87839470616627930192774432062, −4.09871810431929425859815535754, −2.95879238743270528555018245862, −2.13120608483178271133744792033, −0.04433600090585198028911897658, 1.90361760399340678017872101470, 2.68577471480097586988978008462, 3.81743120077881970134876894084, 4.91907273268940123304276597311, 6.09517717564267759936543362205, 7.05498403078476281220936551680, 7.74709153313520121921159218536, 8.567873391370454154258763331809, 9.369550680502853140188373197116, 10.12704376639368704044380474828

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