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Properties

Label 2-1350-5.4-c3-0-21
Degree $2$
Conductor $1350$
Sign $0.447 - 0.894i$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s + 30.5i·7-s + 8i·8-s + 13.5·11-s + 28.0i·13-s + 61.0·14-s + 16·16-s − 55.5i·17-s − 27.4·19-s − 27.0i·22-s − 139. i·23-s + 56.1·26-s − 122. i·28-s + 178.·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.64i·7-s + 0.353i·8-s + 0.371·11-s + 0.598i·13-s + 1.16·14-s + 0.250·16-s − 0.792i·17-s − 0.331·19-s − 0.262i·22-s − 1.26i·23-s + 0.423·26-s − 0.824i·28-s + 1.14·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.597111317\)
\(L(\frac12)\) \(\approx\) \(1.597111317\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 30.5iT - 343T^{2} \)
11 \( 1 - 13.5T + 1.33e3T^{2} \)
13 \( 1 - 28.0iT - 2.19e3T^{2} \)
17 \( 1 + 55.5iT - 4.91e3T^{2} \)
19 \( 1 + 27.4T + 6.85e3T^{2} \)
23 \( 1 + 139. iT - 1.21e4T^{2} \)
29 \( 1 - 178.T + 2.43e4T^{2} \)
31 \( 1 - 297.T + 2.97e4T^{2} \)
37 \( 1 - 159. iT - 5.06e4T^{2} \)
41 \( 1 - 140.T + 6.89e4T^{2} \)
43 \( 1 - 5.68iT - 7.95e4T^{2} \)
47 \( 1 - 301. iT - 1.03e5T^{2} \)
53 \( 1 + 122. iT - 1.48e5T^{2} \)
59 \( 1 + 864.T + 2.05e5T^{2} \)
61 \( 1 + 47.6T + 2.26e5T^{2} \)
67 \( 1 - 402. iT - 3.00e5T^{2} \)
71 \( 1 - 927.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 812.T + 4.93e5T^{2} \)
83 \( 1 - 1.38e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + 124. iT - 9.12e5T^{2} \)
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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

−9.414748682147728531454511180109, −8.660517988272816931335399810078, −8.176149585210469790539796208312, −6.71148630550943429114019056254, −6.08243671316536853724816550295, −4.98085640457811953965922886171, −4.33098216813744037434418802622, −2.85205671214854952425656185500, −2.42793588831242688516741478156, −1.10271328391199555220210371974, 0.42320513906007577364256908561, 1.42051145464607187768061029117, 3.20921362662727138148822446006, 4.08526124839224354621411854239, 4.78243167355855273930496848019, 5.98588461216421299500132934708, 6.65178559811378312509060454824, 7.53234550639368314353757747273, 8.021796999962232571565362775687, 8.974607036114580543571536097291

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