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Properties

Label 2-1350-5.4-c3-0-54
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 − 23i·7-s − 8i·8-s + 30·11-s − 34i·13-s + 46·14-s + 16·16-s + 42i·17-s + 139·19-s + 60i·22-s + 192i·23-s + 68·26-s + 92i·28-s − 234·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.24i·7-s − 0.353i·8-s + 0.822·11-s − 0.725i·13-s + 0.878·14-s + 0.250·16-s + 0.599i·17-s + 1.67·19-s + 0.581i·22-s + 1.74i·23-s + 0.512·26-s + 0.620i·28-s − 1.49·29-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

\(L(2)\) \(\approx\) \(1.948077068\)
\(L(\frac12)\) \(\approx\) \(1.948077068\)
\(L(\frac{5}{2})\) 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 \( 1 + 23iT - 343T^{2} \)
11 \( 1 - 30T + 1.33e3T^{2} \)
13 \( 1 + 34iT - 2.19e3T^{2} \)
17 \( 1 - 42iT - 4.91e3T^{2} \)
19 \( 1 - 139T + 6.85e3T^{2} \)
23 \( 1 - 192iT - 1.21e4T^{2} \)
29 \( 1 + 234T + 2.43e4T^{2} \)
31 \( 1 + 55T + 2.97e4T^{2} \)
37 \( 1 + 191iT - 5.06e4T^{2} \)
41 \( 1 - 138T + 6.89e4T^{2} \)
43 \( 1 - 53iT - 7.95e4T^{2} \)
47 \( 1 + 366iT - 1.03e5T^{2} \)
53 \( 1 + 330iT - 1.48e5T^{2} \)
59 \( 1 - 396T + 2.05e5T^{2} \)
61 \( 1 - 23T + 2.26e5T^{2} \)
67 \( 1 + 452iT - 3.00e5T^{2} \)
71 \( 1 - 204T + 3.57e5T^{2} \)
73 \( 1 + 691iT - 3.89e5T^{2} \)
79 \( 1 - 709T + 4.93e5T^{2} \)
83 \( 1 - 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 - 816T + 7.04e5T^{2} \)
97 \( 1 + 905iT - 9.12e5T^{2} \)
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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

−9.333771564473443982250173851948, −8.089315436676468617745924990323, −7.45412282331189108906789588007, −6.95858885433513371841813469362, −5.79654734428314153900682027796, −5.19142333856758056245891650680, −3.84046592406864886065674703122, −3.52749027710271547491704356211, −1.56476808537157535037200275645, −0.54816388286798262476736766119, 0.996711207653808070261584069649, 2.14823276783165838146648620826, 2.98233355757029309152521709874, 4.06120804337144436867189475944, 5.03118314197903679268832560075, 5.84408649574187243833065410715, 6.79520705165701299812025387110, 7.80517189333905903915727446340, 8.918711807810749363964605666697, 9.198332325488581750882798452470

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