↑

Properties

Label 4-792e2-1.1-c1e2-0-13
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 4·16-s − 5·25-s + 18·31-s − 12·37-s − 2·49-s − 8·64-s + 16·67-s + 14·97-s + 10·100-s − 11·121-s − 36·124-s + 127-s + 131-s + 137-s + 139-s + 24·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 16-s − 25-s + 3.23·31-s − 1.97·37-s − 2/7·49-s − 64-s + 1.95·67-s + 1.42·97-s + 100-s − 121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-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 + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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

\(L(1)\) \(\approx\) \(1.257531159\)
\(L(\frac12)\) \(\approx\) \(1.257531159\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.5.a_f
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.31.as_fj
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.m_dh
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.61.a_cw
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.aq_hh
71$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.71.a_ct
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.ao_dv
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

−8.297858423351776486970693477356, −8.106808707805148257079498368224, −7.63941293516475222320202121138, −7.02927616764016986640786523396, −6.47536762303448627036528108419, −6.17939050885509418696344528358, −5.53574599634031313993894213502, −5.02214215515993588770461846303, −4.74128437563752599818078623862, −4.07875072482213286208485452201, −3.68099919773666417971605530724, −3.05431064544582565016184653726, −2.38371070597652097769642551029, −1.50010531288559013413049148237, −0.60243641146328273498954499795, 0.60243641146328273498954499795, 1.50010531288559013413049148237, 2.38371070597652097769642551029, 3.05431064544582565016184653726, 3.68099919773666417971605530724, 4.07875072482213286208485452201, 4.74128437563752599818078623862, 5.02214215515993588770461846303, 5.53574599634031313993894213502, 6.17939050885509418696344528358, 6.47536762303448627036528108419, 7.02927616764016986640786523396, 7.63941293516475222320202121138, 8.106808707805148257079498368224, 8.297858423351776486970693477356

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