Properties

Label 2-54450-1.1-c1-0-134
Degree $2$
Conductor $54450$
Sign $-1$
Analytic cond. $434.785$
Root an. cond. $20.8515$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3·7-s + 8-s − 4·13-s − 3·14-s + 16-s + 3·17-s + 5·19-s − 4·23-s − 4·26-s − 3·28-s + 5·29-s + 7·31-s + 32-s + 3·34-s − 7·37-s + 5·38-s − 8·41-s + 6·43-s − 4·46-s − 8·47-s + 2·49-s − 4·52-s − 9·53-s − 3·56-s + 5·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 1.10·13-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s − 0.834·23-s − 0.784·26-s − 0.566·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s + 0.514·34-s − 1.15·37-s + 0.811·38-s − 1.24·41-s + 0.914·43-s − 0.589·46-s − 1.16·47-s + 2/7·49-s − 0.554·52-s − 1.23·53-s − 0.400·56-s + 0.656·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(434.785\)
Root analytic conductor: \(20.8515\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 54450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \) 1.7.d
13 \( 1 + 4 T + p T^{2} \) 1.13.e
17 \( 1 - 3 T + p T^{2} \) 1.17.ad
19 \( 1 - 5 T + p T^{2} \) 1.19.af
23 \( 1 + 4 T + p T^{2} \) 1.23.e
29 \( 1 - 5 T + p T^{2} \) 1.29.af
31 \( 1 - 7 T + p T^{2} \) 1.31.ah
37 \( 1 + 7 T + p T^{2} \) 1.37.h
41 \( 1 + 8 T + p T^{2} \) 1.41.i
43 \( 1 - 6 T + p T^{2} \) 1.43.ag
47 \( 1 + 8 T + p T^{2} \) 1.47.i
53 \( 1 + 9 T + p T^{2} \) 1.53.j
59 \( 1 + p T^{2} \) 1.59.a
61 \( 1 - 13 T + p T^{2} \) 1.61.an
67 \( 1 + 12 T + p T^{2} \) 1.67.m
71 \( 1 - 3 T + p T^{2} \) 1.71.ad
73 \( 1 - 6 T + p T^{2} \) 1.73.ag
79 \( 1 + p T^{2} \) 1.79.a
83 \( 1 - 4 T + p T^{2} \) 1.83.ae
89 \( 1 - 15 T + p T^{2} \) 1.89.ap
97 \( 1 + 12 T + p T^{2} \) 1.97.m
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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

−14.48769654780839, −14.18635660053701, −13.63624853186846, −13.25702036704339, −12.49880395237466, −12.21130253749514, −11.88533135606902, −11.28365076675679, −10.40420887067824, −9.979281270731358, −9.796757107310993, −9.101478804614933, −8.273993421393287, −7.800826229105563, −7.208528099469020, −6.598230698270335, −6.292609782757892, −5.495308069228840, −5.042338763367052, −4.494563185722894, −3.615874631480411, −3.209854864730566, −2.701229968275976, −1.905786827475247, −0.9683303596213926, 0, 0.9683303596213926, 1.905786827475247, 2.701229968275976, 3.209854864730566, 3.615874631480411, 4.494563185722894, 5.042338763367052, 5.495308069228840, 6.292609782757892, 6.598230698270335, 7.208528099469020, 7.800826229105563, 8.273993421393287, 9.101478804614933, 9.796757107310993, 9.979281270731358, 10.40420887067824, 11.28365076675679, 11.88533135606902, 12.21130253749514, 12.49880395237466, 13.25702036704339, 13.63624853186846, 14.18635660053701, 14.48769654780839

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