Properties

Label 2-129360-1.1-c1-0-29
Degree $2$
Conductor $129360$
Sign $1$
Analytic cond. $1032.94$
Root an. cond. $32.1394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 11-s + 6·13-s − 15-s + 7·17-s − 5·19-s + 23-s + 25-s + 27-s − 5·29-s − 8·31-s − 33-s − 2·37-s + 6·39-s − 12·41-s + 11·43-s − 45-s + 8·47-s + 7·51-s − 11·53-s + 55-s − 5·57-s − 5·59-s − 7·61-s − 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.258·15-s + 1.69·17-s − 1.14·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.960·39-s − 1.87·41-s + 1.67·43-s − 0.149·45-s + 1.16·47-s + 0.980·51-s − 1.51·53-s + 0.134·55-s − 0.662·57-s − 0.650·59-s − 0.896·61-s − 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1032.94\)
Root analytic conductor: \(32.1394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.507244859\)
\(L(\frac12)\) \(\approx\) \(2.507244859\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−13.49036291968107, −13.00537149855595, −12.52629793959071, −12.25712807975913, −11.47603110599423, −10.98347332953436, −10.66976353487557, −10.18686959161175, −9.506729744064677, −8.952182529732509, −8.674199732156838, −8.080220267662507, −7.632605235752031, −7.253070018384332, −6.550021632477032, −5.877090818357483, −5.633350550366766, −4.842626164981107, −4.163537592087087, −3.654336597711694, −3.334729559712060, −2.697499140866580, −1.717479414344698, −1.441741476742168, −0.4584973356715810, 0.4584973356715810, 1.441741476742168, 1.717479414344698, 2.697499140866580, 3.334729559712060, 3.654336597711694, 4.163537592087087, 4.842626164981107, 5.633350550366766, 5.877090818357483, 6.550021632477032, 7.253070018384332, 7.632605235752031, 8.080220267662507, 8.674199732156838, 8.952182529732509, 9.506729744064677, 10.18686959161175, 10.66976353487557, 10.98347332953436, 11.47603110599423, 12.25712807975913, 12.52629793959071, 13.00537149855595, 13.49036291968107

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