Properties

Label 2-54-9.7-c5-0-3
Degree $2$
Conductor $54$
Sign $-0.0177 + 0.999i$
Analytic cond. $8.66072$
Root an. cond. $2.94291$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (−7.99 − 13.8i)4-s + (20.8 + 36.0i)5-s + (101. − 176. i)7-s − 63.9·8-s + 166.·10-s + (235. − 407. i)11-s + (−241. − 418. i)13-s + (−406. − 704. i)14-s + (−128 + 221. i)16-s − 1.25e3·17-s + 1.97e3·19-s + (332. − 576. i)20-s + (−940. − 1.62e3i)22-s + (239. + 414. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.372 + 0.644i)5-s + (0.784 − 1.35i)7-s − 0.353·8-s + 0.526·10-s + (0.585 − 1.01i)11-s + (−0.396 − 0.686i)13-s + (−0.554 − 0.960i)14-s + (−0.125 + 0.216i)16-s − 1.05·17-s + 1.25·19-s + (0.186 − 0.322i)20-s + (−0.414 − 0.717i)22-s + (0.0942 + 0.163i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $-0.0177 + 0.999i$
Analytic conductor: \(8.66072\)
Root analytic conductor: \(2.94291\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :5/2),\ -0.0177 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.49602 - 1.52283i\)
\(L(\frac12)\) \(\approx\) \(1.49602 - 1.52283i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 3.46i)T \)
3 \( 1 \)
good5 \( 1 + (-20.8 - 36.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-101. + 176. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-235. + 407. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (241. + 418. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.25e3T + 1.41e6T^{2} \)
19 \( 1 - 1.97e3T + 2.47e6T^{2} \)
23 \( 1 + (-239. - 414. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (580. - 1.00e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-1.18e3 - 2.05e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 8.18e3T + 6.93e7T^{2} \)
41 \( 1 + (-8.75e3 - 1.51e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (1.14e4 - 1.98e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (8.68e3 - 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 5.39e3T + 4.18e8T^{2} \)
59 \( 1 + (2.22e4 + 3.85e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.08e3 - 3.60e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.22e3 + 2.12e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.18e3T + 1.80e9T^{2} \)
73 \( 1 - 3.03e3T + 2.07e9T^{2} \)
79 \( 1 + (-2.52e4 + 4.37e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.59e4 - 4.48e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 2.01e4T + 5.58e9T^{2} \)
97 \( 1 + (4.02e4 - 6.96e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−14.01806079270398748107322914004, −13.14921076301728949283136027235, −11.42096089294141322236293671300, −10.83043202382719744689157915034, −9.649955327722189662168692410381, −7.896115601941165256036092242544, −6.41266054615597715118500828673, −4.67365731859264296372406252038, −3.11828557792111657709099390349, −1.06317331775616167608535234690, 2.04268684842429992877001157765, 4.56226860576090330867967049325, 5.58125764953593959094938035481, 7.14485915373132959081428419479, 8.711473895047508031515137170859, 9.449349616822797478264959130797, 11.61810008474262769578267397388, 12.35268965910926244305328206603, 13.62943433658153801900203024180, 14.80133531577249939733919007397

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