Properties

Label 2-5400-5.4-c1-0-21
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  + 2·11-s + 3i·17-s + 19-s − 3i·23-s − 4·29-s − 5·31-s + 10i·37-s + 6·41-s + 6i·43-s + 8i·47-s + 7·49-s − 3i·53-s + 5·61-s − 2i·67-s + 2·71-s + ⋯
L(s)  = 1  + 0.603·11-s + 0.727i·17-s + 0.229·19-s − 0.625i·23-s − 0.742·29-s − 0.898·31-s + 1.64i·37-s + 0.937·41-s + 0.914i·43-s + 1.16i·47-s + 49-s − 0.412i·53-s + 0.640·61-s − 0.244i·67-s + 0.237·71-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

\(L(1)\) \(\approx\) \(1.715571311\)
\(L(\frac12)\) \(\approx\) \(1.715571311\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.225581177931808512771815524367, −7.67419059854830156854979116121, −6.78405464030845245417259376330, −6.21740941516161929256826489364, −5.46513242743369861100795806153, −4.56320977329341254096274747923, −3.87652428614993324478657608781, −3.03506026265434456505779216681, −2.00252257611571403326738883369, −1.02795872785757080851385569456, 0.51194152760306697267346069245, 1.72382265634785801797966192592, 2.62506064172105849044173494636, 3.70717818248219795825376849036, 4.16571129256246658456308924274, 5.44899571892131381918563815503, 5.60246499593008097503309344670, 6.84611206779404725206113332739, 7.21885384900375487185320826446, 7.976411683820534792778389981786

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