Properties

Label 2-2736-1.1-c1-0-36
Degree $2$
Conductor $2736$
Sign $-1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + 7-s − 6·11-s + 5·13-s − 3·17-s − 19-s + 3·23-s − 5·25-s − 9·29-s + 4·31-s + 2·37-s − 8·43-s − 6·49-s + 3·53-s + 9·59-s − 10·61-s − 5·67-s − 6·71-s − 7·73-s − 6·77-s + 10·79-s − 6·83-s + 12·89-s + 5·91-s − 10·97-s − 18·101-s − 14·103-s − 9·107-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.80·11-s + 1.38·13-s − 0.727·17-s − 0.229·19-s + 0.625·23-s − 25-s − 1.67·29-s + 0.718·31-s + 0.328·37-s − 1.21·43-s − 6/7·49-s + 0.412·53-s + 1.17·59-s − 1.28·61-s − 0.610·67-s − 0.712·71-s − 0.819·73-s − 0.683·77-s + 1.12·79-s − 0.658·83-s + 1.27·89-s + 0.524·91-s − 1.01·97-s − 1.79·101-s − 1.37·103-s − 0.870·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
19 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.297610084688077297742296971210, −7.87841813427368117647214734878, −6.98043076884761860510728876245, −6.02372043714400021098795022997, −5.38865185009354292107305454716, −4.54275466905147962254344391783, −3.59163897364434449988676374845, −2.60234956115962506172119076503, −1.59940757777111675726084768164, 0, 1.59940757777111675726084768164, 2.60234956115962506172119076503, 3.59163897364434449988676374845, 4.54275466905147962254344391783, 5.38865185009354292107305454716, 6.02372043714400021098795022997, 6.98043076884761860510728876245, 7.87841813427368117647214734878, 8.297610084688077297742296971210

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