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Properties

Label 2-30e2-20.19-c2-0-51
Degree $2$
Conductor $900$
Sign $0.544 + 0.838i$
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  + (−1.49 − 1.33i)2-s + (0.446 + 3.97i)4-s + 6.56·7-s + (4.63 − 6.52i)8-s − 2.26i·11-s + 14.8i·13-s + (−9.79 − 8.75i)14-s + (−15.6 + 3.55i)16-s − 26.8i·17-s − 10.8i·19-s + (−3.02 + 3.38i)22-s + 36.4·23-s + (19.8 − 22.1i)26-s + (2.93 + 26.1i)28-s − 35.2·29-s + ⋯
L(s)  = 1  + (−0.745 − 0.666i)2-s + (0.111 + 0.993i)4-s + 0.938·7-s + (0.579 − 0.815i)8-s − 0.206i·11-s + 1.14i·13-s + (−0.699 − 0.625i)14-s + (−0.975 + 0.221i)16-s − 1.57i·17-s − 0.572i·19-s + (−0.137 + 0.153i)22-s + 1.58·23-s + (0.762 − 0.853i)26-s + (0.104 + 0.932i)28-s − 1.21·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.544 + 0.838i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.422125254\)
\(L(\frac12)\) \(\approx\) \(1.422125254\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.49 + 1.33i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 6.56T + 49T^{2} \)
11 \( 1 + 2.26iT - 121T^{2} \)
13 \( 1 - 14.8iT - 169T^{2} \)
17 \( 1 + 26.8iT - 289T^{2} \)
19 \( 1 + 10.8iT - 361T^{2} \)
23 \( 1 - 36.4T + 529T^{2} \)
29 \( 1 + 35.2T + 841T^{2} \)
31 \( 1 + 23.8iT - 961T^{2} \)
37 \( 1 - 54.7iT - 1.36e3T^{2} \)
41 \( 1 - 23.8T + 1.68e3T^{2} \)
43 \( 1 - 56.2T + 1.84e3T^{2} \)
47 \( 1 + 51.4T + 2.20e3T^{2} \)
53 \( 1 - 30.6iT - 2.80e3T^{2} \)
59 \( 1 + 6.92iT - 3.48e3T^{2} \)
61 \( 1 - 107.T + 3.72e3T^{2} \)
67 \( 1 - 111.T + 4.48e3T^{2} \)
71 \( 1 - 31.3iT - 5.04e3T^{2} \)
73 \( 1 + 110. iT - 5.32e3T^{2} \)
79 \( 1 + 59.0iT - 6.24e3T^{2} \)
83 \( 1 - 142.T + 6.88e3T^{2} \)
89 \( 1 - 7.14T + 7.92e3T^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−9.473838187544885238799187861939, −9.241238167478614686280736115646, −8.229302929546442027463139662164, −7.37974670958491043142942201559, −6.70756769392478660341152446964, −5.11497792076086435874201828345, −4.34836827146137364121749869619, −3.05155888837909131303638310555, −2.00720426535947201796442250677, −0.77797889258918072502747193559, 1.00273781365372283340967135486, 2.13022428386586771735823931556, 3.80542326935962922816559450961, 5.12591820093048134041190846843, 5.67187916193803758843732452738, 6.77947942593167398535817339245, 7.72635905168747908288379390815, 8.227992619781870164829877375500, 9.042263920749409784272889278677, 9.987387151507787248075586122046

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