Properties

Label 4-1332e2-1.1-c1e2-0-6
Degree $4$
Conductor $1774224$
Sign $1$
Analytic cond. $113.126$
Root an. cond. $3.26129$
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  + 6·5-s − 7-s − 12·11-s + 3·13-s − 12·17-s + 6·19-s + 19·25-s − 6·35-s − 10·37-s + 12·41-s − 24·47-s + 7·49-s + 6·53-s − 72·55-s + 6·59-s + 12·61-s + 18·65-s + 5·67-s + 14·73-s + 12·77-s + 3·79-s + 12·83-s − 72·85-s − 18·89-s − 3·91-s + 36·95-s − 21·109-s + ⋯
L(s)  = 1  + 2.68·5-s − 0.377·7-s − 3.61·11-s + 0.832·13-s − 2.91·17-s + 1.37·19-s + 19/5·25-s − 1.01·35-s − 1.64·37-s + 1.87·41-s − 3.50·47-s + 49-s + 0.824·53-s − 9.70·55-s + 0.781·59-s + 1.53·61-s + 2.23·65-s + 0.610·67-s + 1.63·73-s + 1.36·77-s + 0.337·79-s + 1.31·83-s − 7.80·85-s − 1.90·89-s − 0.314·91-s + 3.69·95-s − 2.01·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1774224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1774224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1774224\)    =    \(2^{4} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(113.126\)
Root analytic conductor: \(3.26129\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1774224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157598043\)
\(L(\frac12)\) \(\approx\) \(2.157598043\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.5.ag_r
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ad_q
17$C_2^2$ \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cn
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ag_bf
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dz
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ct
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.am_ef
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2^2$ \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_de
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_hp
97$C_2^2$ \( 1 - 191 T^{2} + p^{2} T^{4} \) 2.97.a_ahj
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

−9.989027815712174465727264924053, −9.601514581235068858591060583468, −9.117080672245857976845239242100, −8.577989177628591583292633141466, −8.384362691738263242793166804783, −7.901773646611723802970746120283, −7.26128278000413378372406900213, −6.79454192223965092061557930258, −6.56806223312539988361369026567, −5.95908007854455860808213595867, −5.62721942520925047594118687650, −5.18948788014104941735502270143, −5.12968150667544495716855607748, −4.54535514719877011964644927438, −3.61690768228872792689782712373, −2.88023263124936411377127894985, −2.57951132353270871572217226004, −2.11775527186064073100278657050, −1.84729883422493415750899819092, −0.55581628624613320333762480159, 0.55581628624613320333762480159, 1.84729883422493415750899819092, 2.11775527186064073100278657050, 2.57951132353270871572217226004, 2.88023263124936411377127894985, 3.61690768228872792689782712373, 4.54535514719877011964644927438, 5.12968150667544495716855607748, 5.18948788014104941735502270143, 5.62721942520925047594118687650, 5.95908007854455860808213595867, 6.56806223312539988361369026567, 6.79454192223965092061557930258, 7.26128278000413378372406900213, 7.901773646611723802970746120283, 8.384362691738263242793166804783, 8.577989177628591583292633141466, 9.117080672245857976845239242100, 9.601514581235068858591060583468, 9.989027815712174465727264924053

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