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Properties

Label 4-756e2-1.1-c1e2-0-6
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  + 2·5-s − 5·7-s + 4·11-s − 3·13-s + 7·17-s − 5·19-s + 4·23-s + 5·25-s − 29-s − 6·31-s − 10·35-s − 11·37-s − 9·41-s − 5·43-s − 6·47-s + 18·49-s + 3·53-s + 8·55-s + 14·59-s + 6·61-s − 6·65-s + 26·67-s + 16·71-s − 7·73-s − 20·77-s − 18·79-s + 83-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.88·7-s + 1.20·11-s − 0.832·13-s + 1.69·17-s − 1.14·19-s + 0.834·23-s + 25-s − 0.185·29-s − 1.07·31-s − 1.69·35-s − 1.80·37-s − 1.40·41-s − 0.762·43-s − 0.875·47-s + 18/7·49-s + 0.412·53-s + 1.07·55-s + 1.82·59-s + 0.768·61-s − 0.744·65-s + 3.17·67-s + 1.89·71-s − 0.819·73-s − 2.27·77-s − 2.02·79-s + 0.109·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.670838374\)
\(L(\frac12)\) \(\approx\) \(1.670838374\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_ae
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bg
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_abc
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.f_as
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.59.ao_gl
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.61.ag_fb
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.67.aba_lr
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.h_ay
79$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.79.s_jf
83$C_2^2$ \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) 2.83.ab_ade
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.97.ar_hk
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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

−10.31313144760867943305963275787, −9.928632641684825750663932613682, −9.871751115675098161112653304834, −9.418319934230174856377597976382, −8.879464896301950656724252178718, −8.643162652326933815121367626502, −8.103468648819423980249647960001, −7.23101791822365787235798964611, −6.87302740285976810536360452389, −6.73584013999702334254314109400, −6.32589036732423119887338415682, −5.47343146164123109187481022185, −5.47038022724303532255153842673, −4.83102811244788963470027481560, −3.81509550361383763978718273989, −3.57689741879840005906462568462, −3.14894096657691983778769241616, −2.32836138723172585613220795480, −1.72881084394999007448461093778, −0.66143563828744786385086072334, 0.66143563828744786385086072334, 1.72881084394999007448461093778, 2.32836138723172585613220795480, 3.14894096657691983778769241616, 3.57689741879840005906462568462, 3.81509550361383763978718273989, 4.83102811244788963470027481560, 5.47038022724303532255153842673, 5.47343146164123109187481022185, 6.32589036732423119887338415682, 6.73584013999702334254314109400, 6.87302740285976810536360452389, 7.23101791822365787235798964611, 8.103468648819423980249647960001, 8.643162652326933815121367626502, 8.879464896301950656724252178718, 9.418319934230174856377597976382, 9.871751115675098161112653304834, 9.928632641684825750663932613682, 10.31313144760867943305963275787

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