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Properties

Label 4-60e4-1.1-c1e2-0-19
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $826.340$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·11-s + 10·19-s − 4·31-s + 6·41-s + 10·49-s + 4·61-s + 24·71-s − 20·79-s + 30·89-s + 36·101-s + 20·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1.80·11-s + 2.29·19-s − 0.718·31-s + 0.937·41-s + 10/7·49-s + 0.512·61-s + 2.84·71-s − 2.25·79-s + 3.17·89-s + 3.58·101-s + 1.91·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(826.340\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.658584102\)
\(L(\frac12)\) \(\approx\) \(2.658584102\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.67.a_bj
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.83.a_adh
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.589137796847909143080721868858, −8.453675128008026264056927023081, −7.76561490693609783976654679509, −7.68638636784890333635413854636, −7.26173247581483809731111499982, −7.21752622203494613406609226643, −6.43112939428577969932200477043, −6.08563354775183968873758104986, −5.62575007047956842880667036897, −5.30154898817781237797536100529, −5.10028792962733106596879404479, −4.67411892125176430829076338546, −4.08060071447092274663169061553, −3.61294930196898053259364605203, −3.07744435427766439870306567576, −2.95957191182816646295812758825, −2.08586742798045958010227594424, −2.05821291516227602892654943336, −0.895265530485117890000205121204, −0.62261230582933946091806861167, 0.62261230582933946091806861167, 0.895265530485117890000205121204, 2.05821291516227602892654943336, 2.08586742798045958010227594424, 2.95957191182816646295812758825, 3.07744435427766439870306567576, 3.61294930196898053259364605203, 4.08060071447092274663169061553, 4.67411892125176430829076338546, 5.10028792962733106596879404479, 5.30154898817781237797536100529, 5.62575007047956842880667036897, 6.08563354775183968873758104986, 6.43112939428577969932200477043, 7.21752622203494613406609226643, 7.26173247581483809731111499982, 7.68638636784890333635413854636, 7.76561490693609783976654679509, 8.453675128008026264056927023081, 8.589137796847909143080721868858

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