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Properties

Label 4-4600e2-1.1-c1e2-0-12
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s − 12·19-s − 18·29-s + 6·31-s + 6·41-s + 10·49-s − 8·59-s − 20·61-s + 14·71-s + 12·79-s − 32·89-s + 12·101-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 36·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 9-s − 2.75·19-s − 3.34·29-s + 1.07·31-s + 0.937·41-s + 10/7·49-s − 1.04·59-s − 2.56·61-s + 1.66·71-s + 1.35·79-s − 3.39·89-s + 1.19·101-s − 2·121-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 + 1/13·169-s + 2.75·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.3.a_d
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.13.a_ab
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.31.ag_ct
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.73.a_acn
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.83.a_be
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
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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

−8.468214902052962816709798002654, −7.77995806706178982265152148102, −7.46878276599675555428658529202, −7.13302708999685900505024393488, −6.45117348550277483917142938497, −6.43266421744379555071323253550, −5.86510255429341595957904266917, −5.71967753800840004977294169476, −5.29841655185222137949397838535, −4.70519233851939142162232773833, −4.36506737364489301577816132875, −3.94898349350870437470422729977, −3.71767671863564346959783769446, −3.00876975930645899658365000674, −2.68444969132555361863695726691, −2.01664834644638459718230755768, −1.97881782966950329025002164922, −1.08831605253151223855013351164, 0, 0, 1.08831605253151223855013351164, 1.97881782966950329025002164922, 2.01664834644638459718230755768, 2.68444969132555361863695726691, 3.00876975930645899658365000674, 3.71767671863564346959783769446, 3.94898349350870437470422729977, 4.36506737364489301577816132875, 4.70519233851939142162232773833, 5.29841655185222137949397838535, 5.71967753800840004977294169476, 5.86510255429341595957904266917, 6.43266421744379555071323253550, 6.45117348550277483917142938497, 7.13302708999685900505024393488, 7.46878276599675555428658529202, 7.77995806706178982265152148102, 8.468214902052962816709798002654

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