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Properties

Label 2-129360-1.1-c1-0-10
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 − 15-s − 5·17-s − 3·19-s + 9·23-s + 25-s − 27-s − 3·29-s − 2·31-s + 33-s + 37-s − 10·41-s − 7·43-s + 45-s − 5·47-s + 5·51-s − 6·53-s − 55-s + 3·57-s + 3·59-s + 8·61-s − 2·67-s − 9·69-s − 7·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.258·15-s − 1.21·17-s − 0.688·19-s + 1.87·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s + 0.164·37-s − 1.56·41-s − 1.06·43-s + 0.149·45-s − 0.729·47-s + 0.700·51-s − 0.824·53-s − 0.134·55-s + 0.397·57-s + 0.390·59-s + 1.02·61-s − 0.244·67-s − 1.08·69-s − 0.830·71-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

\(L(1)\) \(\approx\) \(1.015955401\)
\(L(\frac12)\) \(\approx\) \(1.015955401\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.26310470664031, −13.08804021992855, −12.70398117022629, −11.98723700528710, −11.48405730832106, −10.94551520948420, −10.84418798440308, −10.07042836696101, −9.722791549476152, −9.095066500218924, −8.593351102471828, −8.284138549324559, −7.290044826019899, −7.087736112775939, −6.430618168760787, −6.156529717989985, −5.326358153903813, −4.930545929896065, −4.617916128461969, −3.724546795108409, −3.231647295597018, −2.440980394008045, −1.899588153351449, −1.249422059875440, −0.3189676305757581, 0.3189676305757581, 1.249422059875440, 1.899588153351449, 2.440980394008045, 3.231647295597018, 3.724546795108409, 4.617916128461969, 4.930545929896065, 5.326358153903813, 6.156529717989985, 6.430618168760787, 7.087736112775939, 7.290044826019899, 8.284138549324559, 8.593351102471828, 9.095066500218924, 9.722791549476152, 10.07042836696101, 10.84418798440308, 10.94551520948420, 11.48405730832106, 11.98723700528710, 12.70398117022629, 13.08804021992855, 13.26310470664031

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