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Properties

Label 2-1350-5.4-c3-0-52
Degree $2$
Conductor $1350$
Sign $0.894 + 0.447i$
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 − 18.7i·7-s − 8i·8-s + 39.9·11-s + 33.7i·13-s + 37.4·14-s + 16·16-s − 53.7i·17-s − 91.7·19-s + 79.9i·22-s + 80.7i·23-s − 67.4·26-s + 74.9i·28-s + 141.·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.01i·7-s − 0.353i·8-s + 1.09·11-s + 0.719i·13-s + 0.715·14-s + 0.250·16-s − 0.766i·17-s − 1.10·19-s + 0.775i·22-s + 0.731i·23-s − 0.509·26-s + 0.506i·28-s + 0.904·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 1350,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.772616593\)
\(L(\frac12)\) \(\approx\) \(1.772616593\)
\(L(\frac{5}{2})\) 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 \( 1 + 18.7iT - 343T^{2} \)
11 \( 1 - 39.9T + 1.33e3T^{2} \)
13 \( 1 - 33.7iT - 2.19e3T^{2} \)
17 \( 1 + 53.7iT - 4.91e3T^{2} \)
19 \( 1 + 91.7T + 6.85e3T^{2} \)
23 \( 1 - 80.7iT - 1.21e4T^{2} \)
29 \( 1 - 141.T + 2.43e4T^{2} \)
31 \( 1 - 264.T + 2.97e4T^{2} \)
37 \( 1 + 61.2iT - 5.06e4T^{2} \)
41 \( 1 + 314.T + 6.89e4T^{2} \)
43 \( 1 - 236. iT - 7.95e4T^{2} \)
47 \( 1 + 243. iT - 1.03e5T^{2} \)
53 \( 1 + 191. iT - 1.48e5T^{2} \)
59 \( 1 + 312.T + 2.05e5T^{2} \)
61 \( 1 + 550.T + 2.26e5T^{2} \)
67 \( 1 + 571. iT - 3.00e5T^{2} \)
71 \( 1 + 183.T + 3.57e5T^{2} \)
73 \( 1 - 125. iT - 3.89e5T^{2} \)
79 \( 1 - 429.T + 4.93e5T^{2} \)
83 \( 1 + 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.23e3T + 7.04e5T^{2} \)
97 \( 1 + 1.53e3iT - 9.12e5T^{2} \)
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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

−9.087989995804166777801723460011, −8.330219003321481061117227443633, −7.40048288045173693878247125709, −6.69344725865159783059091255964, −6.19133905324980786214624707487, −4.77815619868926672569748870810, −4.27853896990051822320532808944, −3.26523164680175382394133761021, −1.64765742760763366175526026073, −0.49048287969180263076948054718, 0.980387674485681593196100756578, 2.12808327433382642512348827633, 3.01117993277206295740567038064, 4.08589596767167353429656748214, 4.92019031048231109063077537515, 6.05927677657595672567636355872, 6.56949404968297282952979881336, 8.055041494542328541838247603940, 8.615094297914657508444516792217, 9.233591621345996041428405345006

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