↑

Properties

Label 4-6e8-1.1-c1e2-0-3
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·11-s + 8·13-s − 2·25-s + 4·37-s − 24·47-s + 2·49-s − 30·59-s + 16·61-s + 12·71-s − 22·73-s + 24·83-s + 26·97-s − 6·107-s + 8·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  − 1.80·11-s + 2.21·13-s − 2/5·25-s + 0.657·37-s − 3.50·47-s + 2/7·49-s − 3.90·59-s + 2.04·61-s + 1.42·71-s − 2.57·73-s + 2.63·83-s + 2.63·97-s − 0.580·107-s + 0.766·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649587999\)
\(L(\frac12)\) \(\approx\) \(1.649587999\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.29.a_abu
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.41.a_acd
43$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.43.a_ach
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.59.be_nf
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.67.a_ach
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.79.a_afq
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.89.a_o
97$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.97.aba_nz
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

−10.12270157187922772990330834902, −9.244079133681734161641388893418, −9.211201324844553621160929352931, −8.476238255088389013042723931012, −8.247911169354740540091454800156, −7.83785219100956297340427987029, −7.67348591381124405402416316808, −6.96380024792757083150558977815, −6.40614632488096096568759833373, −6.11964100201384785641236438444, −5.84027167838508301485762090286, −5.09619397639961647996939544456, −4.91675999308331893814742871519, −4.31894829767982971077675507345, −3.65090867216549021549229303654, −3.24939134565181025566404899769, −2.87389874500256481262357676882, −1.99848449014906957936616513231, −1.53795661505061321203269793216, −0.54148550583982276674566827524, 0.54148550583982276674566827524, 1.53795661505061321203269793216, 1.99848449014906957936616513231, 2.87389874500256481262357676882, 3.24939134565181025566404899769, 3.65090867216549021549229303654, 4.31894829767982971077675507345, 4.91675999308331893814742871519, 5.09619397639961647996939544456, 5.84027167838508301485762090286, 6.11964100201384785641236438444, 6.40614632488096096568759833373, 6.96380024792757083150558977815, 7.67348591381124405402416316808, 7.83785219100956297340427987029, 8.247911169354740540091454800156, 8.476238255088389013042723931012, 9.211201324844553621160929352931, 9.244079133681734161641388893418, 10.12270157187922772990330834902

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