Properties

Label 2-1350-5.4-c3-0-0
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 + 13i·7-s + 8i·8-s − 30·11-s − 61i·13-s + 26·14-s + 16·16-s − 12i·17-s + 49·19-s + 60i·22-s + 18i·23-s − 122·26-s − 52i·28-s + 186·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.701i·7-s + 0.353i·8-s − 0.822·11-s − 1.30i·13-s + 0.496·14-s + 0.250·16-s − 0.171i·17-s + 0.591·19-s + 0.581i·22-s + 0.163i·23-s − 0.920·26-s − 0.350i·28-s + 1.19·29-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

\(L(2)\) \(\approx\) \(0.1849161736\)
\(L(\frac12)\) \(\approx\) \(0.1849161736\)
\(L(\frac{5}{2})\) 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 \( 1 - 13iT - 343T^{2} \)
11 \( 1 + 30T + 1.33e3T^{2} \)
13 \( 1 + 61iT - 2.19e3T^{2} \)
17 \( 1 + 12iT - 4.91e3T^{2} \)
19 \( 1 - 49T + 6.85e3T^{2} \)
23 \( 1 - 18iT - 1.21e4T^{2} \)
29 \( 1 - 186T + 2.43e4T^{2} \)
31 \( 1 + 160T + 2.97e4T^{2} \)
37 \( 1 - 91iT - 5.06e4T^{2} \)
41 \( 1 - 378T + 6.89e4T^{2} \)
43 \( 1 + 268iT - 7.95e4T^{2} \)
47 \( 1 + 144iT - 1.03e5T^{2} \)
53 \( 1 - 570iT - 1.48e5T^{2} \)
59 \( 1 + 204T + 2.05e5T^{2} \)
61 \( 1 + 877T + 2.26e5T^{2} \)
67 \( 1 - 187iT - 3.00e5T^{2} \)
71 \( 1 + 606T + 3.57e5T^{2} \)
73 \( 1 - 431iT - 3.89e5T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 102iT - 5.71e5T^{2} \)
89 \( 1 + 984T + 7.04e5T^{2} \)
97 \( 1 - 265iT - 9.12e5T^{2} \)
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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

−9.564761769758543862653475748844, −8.790808247148220346110217456488, −8.007365240798604708540257608186, −7.26340383987558196194507986400, −5.81714220743060773863603325223, −5.39446144541566994538082280386, −4.34518000464995601005167117026, −3.06568724449418138313411360358, −2.57867646435804228174035187787, −1.17967776171013015864471717909, 0.04564443256450501203090249758, 1.40439505451726550675672085872, 2.82723593440276074018956128534, 4.06194061174577514478596781255, 4.71352126708001978741484292084, 5.73182144320499437276733258103, 6.60371642993232618353889771195, 7.35681374912598594798359376045, 7.970677178766323666608083303722, 8.953832419026667716413265868582

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