Properties

Label 2-272734-1.1-c1-0-94
Degree $2$
Conductor $272734$
Sign $1$
Analytic cond. $2177.79$
Root an. cond. $46.6668$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s + 16-s + 6·17-s + 18-s − 6·19-s − 23-s − 2·24-s − 5·25-s + 4·27-s − 10·29-s − 4·31-s + 32-s + 6·34-s + 36-s − 2·37-s − 6·38-s − 10·41-s + 4·43-s − 46-s − 12·47-s − 2·48-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.208·23-s − 0.408·24-s − 25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.973·38-s − 1.56·41-s + 0.609·43-s − 0.147·46-s − 1.75·47-s − 0.288·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272734\)    =    \(2 \cdot 7^{2} \cdot 11^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(2177.79\)
Root analytic conductor: \(46.6668\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{272734} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 272734,\ (\ :1/2),\ 1)\)

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)$
bad2 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.12845693086714, −12.69798506899573, −12.33151104983003, −11.78982432051386, −11.51982011162680, −11.09544230955230, −10.55109604855267, −10.23826872907971, −9.692142981040777, −9.178223400740502, −8.449626779347890, −8.039426890854924, −7.462548807188925, −7.046018586694465, −6.349217559482741, −6.129797902945494, −5.490679877731573, −5.353935454629644, −4.688590350288101, −4.167050343122565, −3.508947723675445, −3.262523558224759, −2.295763197530794, −1.759469748026957, −1.234518943245509, 0, 0, 1.234518943245509, 1.759469748026957, 2.295763197530794, 3.262523558224759, 3.508947723675445, 4.167050343122565, 4.688590350288101, 5.353935454629644, 5.490679877731573, 6.129797902945494, 6.349217559482741, 7.046018586694465, 7.462548807188925, 8.039426890854924, 8.449626779347890, 9.178223400740502, 9.692142981040777, 10.23826872907971, 10.55109604855267, 11.09544230955230, 11.51982011162680, 11.78982432051386, 12.33151104983003, 12.69798506899573, 13.12845693086714

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