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Properties

Label 2-30e2-20.19-c2-0-38
Degree $2$
Conductor $900$
Sign $0.862 + 0.505i$
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  + (−1.36 − 1.46i)2-s + (−0.265 + 3.99i)4-s + 1.87·7-s + (6.19 − 5.06i)8-s + 15.1i·11-s − 18.1i·13-s + (−2.55 − 2.73i)14-s + (−15.8 − 2.11i)16-s + 12.6i·17-s − 28.1i·19-s + (22.0 − 20.6i)22-s + 10.9·23-s + (−26.4 + 24.7i)26-s + (−0.496 + 7.46i)28-s + 9.95·29-s + ⋯
L(s)  = 1  + (−0.683 − 0.730i)2-s + (−0.0663 + 0.997i)4-s + 0.267·7-s + (0.773 − 0.633i)8-s + 1.37i·11-s − 1.39i·13-s + (−0.182 − 0.195i)14-s + (−0.991 − 0.132i)16-s + 0.746i·17-s − 1.48i·19-s + (1.00 − 0.938i)22-s + 0.475·23-s + (−1.01 + 0.952i)26-s + (−0.0177 + 0.266i)28-s + 0.343·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.862 + 0.505i$
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: \((2,\ 900,\ (\ :1),\ 0.862 + 0.505i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.293647087\)
\(L(\frac12)\) \(\approx\) \(1.293647087\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 1.46i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.87T + 49T^{2} \)
11 \( 1 - 15.1iT - 121T^{2} \)
13 \( 1 + 18.1iT - 169T^{2} \)
17 \( 1 - 12.6iT - 289T^{2} \)
19 \( 1 + 28.1iT - 361T^{2} \)
23 \( 1 - 10.9T + 529T^{2} \)
29 \( 1 - 9.95T + 841T^{2} \)
31 \( 1 - 24.4iT - 961T^{2} \)
37 \( 1 - 17.8iT - 1.36e3T^{2} \)
41 \( 1 + 28.8T + 1.68e3T^{2} \)
43 \( 1 - 56.3T + 1.84e3T^{2} \)
47 \( 1 - 49.5T + 2.20e3T^{2} \)
53 \( 1 + 60.1iT - 2.80e3T^{2} \)
59 \( 1 - 110. iT - 3.48e3T^{2} \)
61 \( 1 - 34.2T + 3.72e3T^{2} \)
67 \( 1 - 131.T + 4.48e3T^{2} \)
71 \( 1 - 52.0iT - 5.04e3T^{2} \)
73 \( 1 - 62.2iT - 5.32e3T^{2} \)
79 \( 1 + 103. iT - 6.24e3T^{2} \)
83 \( 1 - 57.2T + 6.88e3T^{2} \)
89 \( 1 - 145.T + 7.92e3T^{2} \)
97 \( 1 + 66.7iT - 9.40e3T^{2} \)
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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

−10.00754398042719089825561109182, −9.061189275774797650314701759939, −8.310947268780838943093756923199, −7.45600329363547692329201378406, −6.74696734707498743353361173304, −5.22174593849946979516768467896, −4.36784060932487570044445359992, −3.13952355024099121150649624726, −2.16025465483065464938660565551, −0.838722909524131129365490169068, 0.791954348508190684516453742887, 2.11071331798509364631734171008, 3.73467286157204713868259984794, 4.89950233957576199832210815740, 5.87347306350685854708685234131, 6.55293660092961271993737568254, 7.55916392120238975194492167234, 8.290343447877075746085152657223, 9.088302838460528921238633863780, 9.703414186811009808819796653435

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