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Properties

Label 2-1350-5.4-c3-0-62
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 − 24.6i·7-s + 8i·8-s + 44.3·11-s − 78.4i·13-s − 49.3·14-s + 16·16-s − 91.4i·17-s + 89.4·19-s − 88.7i·22-s + 82.4i·23-s − 156.·26-s + 98.7i·28-s + 13.4·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 1.33i·7-s + 0.353i·8-s + 1.21·11-s − 1.67i·13-s − 0.942·14-s + 0.250·16-s − 1.30i·17-s + 1.08·19-s − 0.860i·22-s + 0.747i·23-s − 1.18·26-s + 0.666i·28-s + 0.0863·29-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

\(L(2)\) \(\approx\) \(2.300033419\)
\(L(\frac12)\) \(\approx\) \(2.300033419\)
\(L(\frac{5}{2})\) 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 \( 1 + 24.6iT - 343T^{2} \)
11 \( 1 - 44.3T + 1.33e3T^{2} \)
13 \( 1 + 78.4iT - 2.19e3T^{2} \)
17 \( 1 + 91.4iT - 4.91e3T^{2} \)
19 \( 1 - 89.4T + 6.85e3T^{2} \)
23 \( 1 - 82.4iT - 1.21e4T^{2} \)
29 \( 1 - 13.4T + 2.43e4T^{2} \)
31 \( 1 - 268.T + 2.97e4T^{2} \)
37 \( 1 + 249. iT - 5.06e4T^{2} \)
41 \( 1 - 298.T + 6.89e4T^{2} \)
43 \( 1 - 262. iT - 7.95e4T^{2} \)
47 \( 1 - 595. iT - 1.03e5T^{2} \)
53 \( 1 + 457. iT - 1.48e5T^{2} \)
59 \( 1 - 370.T + 2.05e5T^{2} \)
61 \( 1 + 449.T + 2.26e5T^{2} \)
67 \( 1 + 385. iT - 3.00e5T^{2} \)
71 \( 1 - 775.T + 3.57e5T^{2} \)
73 \( 1 - 341. iT - 3.89e5T^{2} \)
79 \( 1 - 371.T + 4.93e5T^{2} \)
83 \( 1 + 1.18e3iT - 5.71e5T^{2} \)
89 \( 1 - 197.T + 7.04e5T^{2} \)
97 \( 1 + 860. iT - 9.12e5T^{2} \)
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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

−9.256833160288142763400420989462, −7.913577674438285140446909574472, −7.50505923926140422011924645444, −6.44621610949989388679932591651, −5.35042021347151860777067518284, −4.45166492744902515222912764982, −3.54788960952889417207526889279, −2.81004333359617770330068213905, −1.12273484100279842013282668729, −0.67262867842147285074309286182, 1.25942731214685309930235415499, 2.38237164041087317044806689321, 3.76796570573306297709800554066, 4.55834201521843506176225235190, 5.60468816503859159126533058151, 6.41036256572874390084073434885, 6.84061488928229194247315920393, 8.130022236705514337354103948755, 8.783509713336793359001065107997, 9.279541820986772902287426773509

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