Properties

Label 4-968e2-1.1-c1e2-0-4
Degree $4$
Conductor $937024$
Sign $1$
Analytic cond. $59.7454$
Root an. cond. $2.78020$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 6·9-s + 4·16-s + 4·23-s + 10·25-s + 12·31-s − 12·36-s + 20·47-s − 14·49-s − 8·64-s − 28·71-s + 27·81-s + 4·89-s − 8·92-s + 12·97-s − 20·100-s + 36·103-s + 20·113-s − 24·124-s + ⋯
L(s)  = 1  − 4-s + 2·9-s + 16-s + 0.834·23-s + 2·25-s + 2.15·31-s − 2·36-s + 2.91·47-s − 2·49-s − 64-s − 3.32·71-s + 3·81-s + 0.423·89-s − 0.834·92-s + 1.21·97-s − 2·100-s + 3.54·103-s + 1.88·113-s − 2.15·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(937024\)    =    \(2^{6} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(59.7454\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 937024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326310319\)
\(L(\frac12)\) \(\approx\) \(2.326310319\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
11 \( 1 \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.37.a_acw
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.43.a_bq
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.47.au_hm
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.61.a_da
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.71.bc_na
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.83.a_es
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−10.17440951487034513174929497875, −10.03238912250516599509600246345, −9.183813115741581049805925055282, −9.131366973068805457111189596295, −8.658489509254787418696662431313, −8.243315972515755923664297534628, −7.52864460826027345346465875768, −7.39007257810625432042603398346, −6.97563918450627328965898162439, −6.25892744006246041898340407936, −6.14315377680111810234821922068, −5.20926167413079623020774110075, −4.82963335950659465988548432771, −4.45302980993295696703411544398, −4.28436435998252042303799546447, −3.40016117896321326539672350353, −3.05718981130427437984995911984, −2.19799297070616868395560475179, −1.19110428123640505085653426766, −0.918643532589292712111048271343, 0.918643532589292712111048271343, 1.19110428123640505085653426766, 2.19799297070616868395560475179, 3.05718981130427437984995911984, 3.40016117896321326539672350353, 4.28436435998252042303799546447, 4.45302980993295696703411544398, 4.82963335950659465988548432771, 5.20926167413079623020774110075, 6.14315377680111810234821922068, 6.25892744006246041898340407936, 6.97563918450627328965898162439, 7.39007257810625432042603398346, 7.52864460826027345346465875768, 8.243315972515755923664297534628, 8.658489509254787418696662431313, 9.131366973068805457111189596295, 9.183813115741581049805925055282, 10.03238912250516599509600246345, 10.17440951487034513174929497875

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