Properties

Label 2-129360-1.1-c1-0-102
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 11-s + 6·13-s + 15-s + 6·17-s − 4·19-s + 4·23-s + 25-s + 27-s + 2·29-s + 33-s + 2·37-s + 6·39-s + 2·41-s + 4·43-s + 45-s − 8·47-s + 6·51-s + 2·53-s + 55-s − 4·57-s + 8·59-s − 2·61-s + 6·65-s − 4·67-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 + 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.174·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 0.840·51-s + 0.274·53-s + 0.134·55-s − 0.529·57-s + 1.04·59-s − 0.256·61-s + 0.744·65-s − 0.488·67-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

\(L(1)\) \(\approx\) \(5.267148457\)
\(L(\frac12)\) \(\approx\) \(5.267148457\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.48800124355140, −12.94075651367017, −12.82810204423756, −12.07972512318788, −11.49029433947874, −11.11844401743028, −10.35753073717160, −10.27553863486574, −9.526603495514988, −9.018854657914226, −8.676124990677798, −8.163151995678047, −7.691250136712011, −7.052184416429515, −6.442362542636935, −6.096401217187915, −5.514464561199143, −4.936444556669994, −4.142098596630311, −3.804503385856754, −3.097824877836402, −2.718162608738545, −1.784227560966127, −1.321205613606950, −0.7132240131358429, 0.7132240131358429, 1.321205613606950, 1.784227560966127, 2.718162608738545, 3.097824877836402, 3.804503385856754, 4.142098596630311, 4.936444556669994, 5.514464561199143, 6.096401217187915, 6.442362542636935, 7.052184416429515, 7.691250136712011, 8.163151995678047, 8.676124990677798, 9.018854657914226, 9.526603495514988, 10.27553863486574, 10.35753073717160, 11.11844401743028, 11.49029433947874, 12.07972512318788, 12.82810204423756, 12.94075651367017, 13.48800124355140

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