Properties

Label 2-1350-5.4-c3-0-12
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 + 22i·7-s + 8i·8-s − 12·11-s + 38i·13-s + 44·14-s + 16·16-s + 105i·17-s + 157·19-s + 24i·22-s − 117i·23-s + 76·26-s − 88i·28-s − 66·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.18i·7-s + 0.353i·8-s − 0.328·11-s + 0.810i·13-s + 0.839·14-s + 0.250·16-s + 1.49i·17-s + 1.89·19-s + 0.232i·22-s − 1.06i·23-s + 0.573·26-s − 0.593i·28-s − 0.422·29-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

\(L(2)\) \(\approx\) \(0.9825646034\)
\(L(\frac12)\) \(\approx\) \(0.9825646034\)
\(L(\frac{5}{2})\) 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 \( 1 - 22iT - 343T^{2} \)
11 \( 1 + 12T + 1.33e3T^{2} \)
13 \( 1 - 38iT - 2.19e3T^{2} \)
17 \( 1 - 105iT - 4.91e3T^{2} \)
19 \( 1 - 157T + 6.85e3T^{2} \)
23 \( 1 + 117iT - 1.21e4T^{2} \)
29 \( 1 + 66T + 2.43e4T^{2} \)
31 \( 1 + 25T + 2.97e4T^{2} \)
37 \( 1 + 314iT - 5.06e4T^{2} \)
41 \( 1 + 504T + 6.89e4T^{2} \)
43 \( 1 - 380iT - 7.95e4T^{2} \)
47 \( 1 - 252iT - 1.03e5T^{2} \)
53 \( 1 - 3iT - 1.48e5T^{2} \)
59 \( 1 - 318T + 2.05e5T^{2} \)
61 \( 1 - 293T + 2.26e5T^{2} \)
67 \( 1 - 322iT - 3.00e5T^{2} \)
71 \( 1 + 120T + 3.57e5T^{2} \)
73 \( 1 - 44iT - 3.89e5T^{2} \)
79 \( 1 + 917T + 4.93e5T^{2} \)
83 \( 1 - 309iT - 5.71e5T^{2} \)
89 \( 1 + 1.27e3T + 7.04e5T^{2} \)
97 \( 1 + 1.32e3iT - 9.12e5T^{2} \)
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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

−9.547309246470279361367649356665, −8.774161673056846870998681420824, −8.181707610513285595457663948353, −7.07933726437958444918465428674, −5.96418774353224261746537223386, −5.33029618947511257544340545158, −4.28655536222325652550279685306, −3.25996529209873327360309095430, −2.31815909939069258153273437822, −1.38319702431786030342843126942, 0.24331252468986436478893736897, 1.22315563001457299801655859828, 3.03730823304192081714980406823, 3.76935632079978045677505634475, 5.13176057874360848695290270392, 5.35507128140237101930150216017, 6.80127324522516639880146112948, 7.34644571316452324760322061060, 7.86477713327128268962068854360, 8.908779307400659211550841679580

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