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Properties

Label 4-2700e2-1.1-c1e2-0-13
Degree $4$
Conductor $7290000$
Sign $1$
Analytic cond. $464.816$
Root an. cond. $4.64323$
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  + 12·11-s + 14·19-s − 14·31-s + 12·41-s − 2·49-s + 24·59-s − 14·61-s + 12·71-s + 2·79-s + 12·89-s + 24·101-s − 10·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 3.61·11-s + 3.21·19-s − 2.51·31-s + 1.87·41-s − 2/7·49-s + 3.12·59-s − 1.79·61-s + 1.42·71-s + 0.225·79-s + 1.27·89-s + 2.38·101-s − 0.957·109-s + 7.81·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7290000\)    =    \(2^{4} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(464.816\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.627011124\)
\(L(\frac12)\) \(\approx\) \(4.627011124\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.19.ao_dj
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.23.a_bj
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.79.ac_gd
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.83.a_adh
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−9.125740913615331245706947465305, −8.982264768566373308306878050506, −8.293081617996365040222772012801, −7.77949216299918700198861894775, −7.41106351860769549429966298850, −7.13265523286790592901950410007, −6.81915472312370480692841830321, −6.41442641159231362790223570674, −5.86698871097936110969219827407, −5.70823624518412527148083558985, −5.15337934341860553954170718559, −4.73324949714790720570673304944, −4.05437890013519934300945849229, −3.81214660276834380752705002785, −3.48234902022798454151523702694, −3.16790386821563528800210065635, −2.23837567287528517544905511816, −1.71699487149532518148958811031, −1.03135822643077417071071172978, −0.949386231486950069077761217962, 0.949386231486950069077761217962, 1.03135822643077417071071172978, 1.71699487149532518148958811031, 2.23837567287528517544905511816, 3.16790386821563528800210065635, 3.48234902022798454151523702694, 3.81214660276834380752705002785, 4.05437890013519934300945849229, 4.73324949714790720570673304944, 5.15337934341860553954170718559, 5.70823624518412527148083558985, 5.86698871097936110969219827407, 6.41442641159231362790223570674, 6.81915472312370480692841830321, 7.13265523286790592901950410007, 7.41106351860769549429966298850, 7.77949216299918700198861894775, 8.293081617996365040222772012801, 8.982264768566373308306878050506, 9.125740913615331245706947465305

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