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Properties

Label 4-792e2-1.1-c1e2-0-18
Degree $4$
Conductor $627264$
Sign $1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·11-s + 2·17-s + 6·25-s − 2·29-s + 12·31-s + 4·37-s − 6·41-s + 2·49-s − 24·67-s − 4·83-s − 8·97-s − 2·101-s + 24·103-s + 4·107-s + 25·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.80·11-s + 0.485·17-s + 6/5·25-s − 0.371·29-s + 2.15·31-s + 0.657·37-s − 0.937·41-s + 2/7·49-s − 2.93·67-s − 0.439·83-s − 0.812·97-s − 0.199·101-s + 2.36·103-s + 0.386·107-s + 2.27·121-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 − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.520950597\)
\(L(\frac12)\) \(\approx\) \(1.520950597\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ac_ba
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.c_bi
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.am_dq
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.g_co
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.43.a_bu
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.53.a_by
59$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.59.a_by
61$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.61.a_g
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.73.a_co
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.e_cs
89$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.89.a_cw
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.i_hy
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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

−8.220360823339759419006551008492, −8.035909903774983823134603452654, −7.51182889491257422369255213093, −7.14743292446675118586268964439, −6.56858247838639680170781105978, −6.06128726386395811021755083698, −5.62663526961907192326610983572, −5.11789155625179626066968508769, −4.64470131448889184216035506328, −4.32119124261040665174484819843, −3.33357816393585859922983279888, −2.92956586559013282292874150295, −2.52987136530612911000240338476, −1.61986729644494460994142458606, −0.62691583743259270301489087147, 0.62691583743259270301489087147, 1.61986729644494460994142458606, 2.52987136530612911000240338476, 2.92956586559013282292874150295, 3.33357816393585859922983279888, 4.32119124261040665174484819843, 4.64470131448889184216035506328, 5.11789155625179626066968508769, 5.62663526961907192326610983572, 6.06128726386395811021755083698, 6.56858247838639680170781105978, 7.14743292446675118586268964439, 7.51182889491257422369255213093, 8.035909903774983823134603452654, 8.220360823339759419006551008492

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