Properties

Label 4-960e2-1.1-c1e2-0-68
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 3·9-s − 4·13-s + 4·15-s − 25-s − 4·27-s − 16·31-s − 20·37-s + 8·39-s − 12·41-s + 8·43-s − 6·45-s + 10·49-s − 20·53-s + 8·65-s − 24·67-s + 2·75-s − 32·79-s + 5·81-s + 8·83-s + 20·89-s + 32·93-s − 40·107-s + 40·111-s − 12·117-s + 18·121-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s + 1.03·15-s − 1/5·25-s − 0.769·27-s − 2.87·31-s − 3.28·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s − 0.894·45-s + 10/7·49-s − 2.74·53-s + 0.992·65-s − 2.93·67-s + 0.230·75-s − 3.60·79-s + 5/9·81-s + 0.878·83-s + 2.11·89-s + 3.31·93-s − 3.86·107-s + 3.79·111-s − 1.10·117-s + 1.63·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.980077244689985219438149759238, −9.377176625104454167169594659837, −8.906336837388073723018677622144, −8.810717149694340002757902011484, −7.84562954254642094580790714719, −7.72064336237029501876259380126, −7.13697693528889812950181903327, −7.01023712489750114353038176992, −6.49257983207198492917992116414, −5.75708564521409225722176714103, −5.37266462380166856013116927107, −5.21712453440728330961629787336, −4.35078820618718110737250047094, −4.26984559009942227425452921018, −3.36440270809281685118976063726, −3.17655374663426220336038061706, −1.92361223104380861903812238233, −1.61332349330505361767442043007, 0, 0, 1.61332349330505361767442043007, 1.92361223104380861903812238233, 3.17655374663426220336038061706, 3.36440270809281685118976063726, 4.26984559009942227425452921018, 4.35078820618718110737250047094, 5.21712453440728330961629787336, 5.37266462380166856013116927107, 5.75708564521409225722176714103, 6.49257983207198492917992116414, 7.01023712489750114353038176992, 7.13697693528889812950181903327, 7.72064336237029501876259380126, 7.84562954254642094580790714719, 8.810717149694340002757902011484, 8.906336837388073723018677622144, 9.377176625104454167169594659837, 9.980077244689985219438149759238

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