Properties

Label 2-41280-1.1-c1-0-77
Degree $2$
Conductor $41280$
Sign $-1$
Analytic cond. $329.622$
Root an. cond. $18.1555$
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  − 3-s + 5-s + 4·7-s + 9-s + 2·13-s − 15-s + 2·17-s − 4·19-s − 4·21-s + 25-s − 27-s − 2·29-s + 4·35-s − 10·37-s − 2·39-s − 6·41-s + 43-s + 45-s − 8·47-s + 9·49-s − 2·51-s − 2·53-s + 4·57-s − 2·61-s + 4·63-s + 2·65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.676·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s − 0.256·61-s + 0.503·63-s + 0.248·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41280\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(329.622\)
Root analytic conductor: \(18.1555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{41280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 41280,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
43 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.00560455281944, −14.44460298040716, −13.99510330041890, −13.56928094552481, −12.78468115390152, −12.48354866574430, −11.73431004649986, −11.32114422242853, −10.94463891452736, −10.31026186114969, −9.974686338928951, −9.094267386022543, −8.571707341802885, −8.148068149065218, −7.540117319706188, −6.840136796634084, −6.343130915155094, −5.646287717916333, −5.130395483648752, −4.750058376078101, −3.970142901279768, −3.348488033180142, −2.271177803245036, −1.690315716129054, −1.175177382089044, 0, 1.175177382089044, 1.690315716129054, 2.271177803245036, 3.348488033180142, 3.970142901279768, 4.750058376078101, 5.130395483648752, 5.646287717916333, 6.343130915155094, 6.840136796634084, 7.540117319706188, 8.148068149065218, 8.571707341802885, 9.094267386022543, 9.974686338928951, 10.31026186114969, 10.94463891452736, 11.32114422242853, 11.73431004649986, 12.48354866574430, 12.78468115390152, 13.56928094552481, 13.99510330041890, 14.44460298040716, 15.00560455281944

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