Properties

Degree $2$
Conductor $177744$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 2·13-s + 2·15-s + 6·17-s + 4·19-s + 21-s − 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s + 2·35-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 6·51-s − 6·53-s − 8·55-s + 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.338·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177744\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{177744} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177744,\ (\ :1/2),\ -1)\)

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

−13.53777539773098, −13.02429954425075, −12.52644175214639, −12.01041757663474, −11.67901644812803, −10.93187397567737, −10.35785255658514, −10.02238165583311, −9.736517161212597, −9.278848136620373, −8.390679772389295, −8.292080990581898, −7.617215949496333, −7.302204424086180, −6.701985354718760, −5.952286089681326, −5.488787612770320, −5.126765741211801, −4.685021334237547, −3.792135813135787, −3.280459495757854, −2.691475374170654, −2.285156528249962, −1.556783053300525, −1.037814448216102, 0, 1.037814448216102, 1.556783053300525, 2.285156528249962, 2.691475374170654, 3.280459495757854, 3.792135813135787, 4.685021334237547, 5.126765741211801, 5.488787612770320, 5.952286089681326, 6.701985354718760, 7.302204424086180, 7.617215949496333, 8.292080990581898, 8.390679772389295, 9.278848136620373, 9.736517161212597, 10.02238165583311, 10.35785255658514, 10.93187397567737, 11.67901644812803, 12.01041757663474, 12.52644175214639, 13.02429954425075, 13.53777539773098

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