Properties

Label 2-5400-5.4-c1-0-18
Degree $2$
Conductor $5400$
Sign $-0.447 - 0.894i$
Analytic cond. $43.1192$
Root an. cond. $6.56652$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·7-s + 4·11-s + i·13-s + 4i·17-s + 19-s + 4i·23-s − 4·31-s + 9i·37-s − 8i·43-s + 12i·47-s − 2·49-s − 8i·53-s − 4·59-s − 5·61-s − 11i·67-s + ⋯
L(s)  = 1  + 1.13i·7-s + 1.20·11-s + 0.277i·13-s + 0.970i·17-s + 0.229·19-s + 0.834i·23-s − 0.718·31-s + 1.47i·37-s − 1.21i·43-s + 1.75i·47-s − 0.285·49-s − 1.09i·53-s − 0.520·59-s − 0.640·61-s − 1.34i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5400\)    =    \(2^{3} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(43.1192\)
Root analytic conductor: \(6.56652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5400} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5400,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.707508734\)
\(L(\frac12)\) \(\approx\) \(1.707508734\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 9iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−8.470360958805363914906805967017, −7.79069649305554665643431146012, −6.84075251870433879531768741192, −6.23677329590089121039487044327, −5.62242918571371554028641927105, −4.80208071255630176286821099423, −3.87509137148009522121856517899, −3.19696358568573385484819134056, −2.07461257374055932086993191387, −1.34570797303715441464810617132, 0.46952281390110531252636695270, 1.37354839828689688535446946434, 2.56688594706573486144994342202, 3.59748960480907163482422588339, 4.14910827577004919924162215779, 4.92855318238906594259106760175, 5.84210667852036916964007291831, 6.66682891041724654249548330039, 7.21496693114564215195257678505, 7.76541664297207504092583255487

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