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Properties

Label 4-1008e2-1.1-c1e2-0-45
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $64.7851$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 16·19-s + 10·25-s − 8·31-s + 20·37-s + 9·49-s + 40·103-s + 4·109-s + 22·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 40·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 1.51·7-s − 3.67·19-s + 2·25-s − 1.43·31-s + 3.28·37-s + 9/7·49-s + 3.94·103-s + 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-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 − 1.69·169-s + 0.0760·173-s + 3.02·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(64.7851\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.229364470\)
\(L(\frac12)\) \(\approx\) \(2.229364470\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
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_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−10.13492454153503060912432204510, −9.974475075147220130899211934786, −9.043254092535217975730650784603, −8.767235446291592073544157948120, −8.730378237228071245125837124866, −8.113415734673425054197306037503, −7.68989240239546328973781794328, −7.41358040040647653967041732677, −6.67274662101227327522764097490, −6.37447343321520701377945717431, −5.99201359655654836576137737500, −5.38956180402213963716455353341, −4.70862159544870010970558261220, −4.52641487914003300339760861398, −4.20043054938959377757651461653, −3.48970516089307351932069181564, −2.57701478862453187354129891355, −2.23353128743320239538718974380, −1.64591557034862902727546244494, −0.69554242727183577572619045318, 0.69554242727183577572619045318, 1.64591557034862902727546244494, 2.23353128743320239538718974380, 2.57701478862453187354129891355, 3.48970516089307351932069181564, 4.20043054938959377757651461653, 4.52641487914003300339760861398, 4.70862159544870010970558261220, 5.38956180402213963716455353341, 5.99201359655654836576137737500, 6.37447343321520701377945717431, 6.67274662101227327522764097490, 7.41358040040647653967041732677, 7.68989240239546328973781794328, 8.113415734673425054197306037503, 8.730378237228071245125837124866, 8.767235446291592073544157948120, 9.043254092535217975730650784603, 9.974475075147220130899211934786, 10.13492454153503060912432204510

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