Properties

Label 4-4608e2-1.1-c1e2-0-19
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  − 4·5-s + 12·13-s + 2·25-s − 4·29-s + 12·37-s − 12·41-s − 6·49-s + 4·53-s + 12·61-s − 48·65-s − 24·73-s + 24·89-s − 16·97-s + 20·101-s − 12·109-s − 12·113-s − 4·121-s + 28·125-s + ⋯
L(s)  = 1  − 1.78·5-s + 3.32·13-s + 2/5·25-s − 0.742·29-s + 1.97·37-s − 1.87·41-s − 6/7·49-s + 0.549·53-s + 1.53·61-s − 5.95·65-s − 2.80·73-s + 2.54·89-s − 1.62·97-s + 1.99·101-s − 1.14·109-s − 1.12·113-s − 0.363·121-s + 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813967491\)
\(L(\frac12)\) \(\approx\) \(1.813967491\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.43.a_cq
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.59.a_em
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.67.a_abc
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.79.a_ew
83$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \) 2.83.a_fs
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−8.349444429624400362971182332018, −8.148464343214914240274222375180, −7.88244548973680187254743208149, −7.58714524218973146144740967845, −7.03869804662483498604701426730, −6.74184113657214989248999355714, −6.21006886609067501810551759796, −6.13011726398830841645193036098, −5.52967668805551152478188874953, −5.35191480352664701037590867759, −4.45226709758704579845417856741, −4.34859853697519784802575647802, −3.80878500304498434124066446376, −3.77925851746322915252236534502, −3.15596263630828736567863014413, −3.06177633958413806739042841663, −2.01476109282332929515480213417, −1.62146359742865250715272264316, −0.982511811595873951636118131319, −0.45806019799875928694739539085, 0.45806019799875928694739539085, 0.982511811595873951636118131319, 1.62146359742865250715272264316, 2.01476109282332929515480213417, 3.06177633958413806739042841663, 3.15596263630828736567863014413, 3.77925851746322915252236534502, 3.80878500304498434124066446376, 4.34859853697519784802575647802, 4.45226709758704579845417856741, 5.35191480352664701037590867759, 5.52967668805551152478188874953, 6.13011726398830841645193036098, 6.21006886609067501810551759796, 6.74184113657214989248999355714, 7.03869804662483498604701426730, 7.58714524218973146144740967845, 7.88244548973680187254743208149, 8.148464343214914240274222375180, 8.349444429624400362971182332018

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