Properties

Label 4-648e2-1.1-c1e2-0-22
Degree $4$
Conductor $419904$
Sign $1$
Analytic cond. $26.7734$
Root an. cond. $2.27471$
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  + 5-s + 3·7-s − 2·11-s + 5·13-s + 5·17-s + 7·19-s − 5·23-s − 25-s + 3·29-s + 7·31-s + 3·35-s + 6·37-s + 12·41-s + 8·43-s − 3·47-s + 49-s + 10·53-s − 2·55-s − 14·59-s − 61-s + 5·65-s + 4·67-s + 8·71-s − 7·73-s − 6·77-s − 7·79-s − 25·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 0.603·11-s + 1.38·13-s + 1.21·17-s + 1.60·19-s − 1.04·23-s − 1/5·25-s + 0.557·29-s + 1.25·31-s + 0.507·35-s + 0.986·37-s + 1.87·41-s + 1.21·43-s − 0.437·47-s + 1/7·49-s + 1.37·53-s − 0.269·55-s − 1.82·59-s − 0.128·61-s + 0.620·65-s + 0.488·67-s + 0.949·71-s − 0.819·73-s − 0.683·77-s − 0.787·79-s − 2.74·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(419904\)    =    \(2^{6} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(26.7734\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 419904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.876594316\)
\(L(\frac12)\) \(\approx\) \(2.876594316\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_c
7$D_{4}$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_i
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$D_{4}$ \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_y
17$D_{4}$ \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_bg
19$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.19.ah_bq
23$D_{4}$ \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bs
29$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_ca
31$D_{4}$ \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.31.ah_co
37$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_by
41$D_{4}$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dh
43$D_{4}$ \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_cr
47$D_{4}$ \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_dk
53$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.53.ak_du
59$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.59.o_gl
61$D_{4}$ \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \) 2.61.b_ek
67$D_{4}$ \( 1 - 4 T + 105 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_eb
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$D_{4}$ \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_dg
79$D_{4}$ \( 1 + 7 T + 162 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.79.h_gg
83$D_{4}$ \( 1 + 25 T + 314 T^{2} + 25 p T^{3} + p^{2} T^{4} \) 2.83.z_mc
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$D_{4}$ \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_gv
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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

−10.72594872634854491426259276227, −10.40320328742561076757660148309, −9.770233097794993171001989060540, −9.681834509538176402191077040757, −9.059708945326420929361000907247, −8.493125287248278832640608416906, −8.005255546238632463989981397215, −7.892261528771830579960686916868, −7.43614633595531194995291118169, −6.79767342090842340018109565393, −5.94005669964667673908676027308, −5.93284244591264969411411532768, −5.42320271397944194786082854955, −4.85807083052575407576807696689, −4.17507071256542274008972909809, −3.86844686585152412526781788985, −2.80282201105271943883491328924, −2.67566887150177351662702175336, −1.38209415952852671197512345307, −1.15981756176195471990440323267, 1.15981756176195471990440323267, 1.38209415952852671197512345307, 2.67566887150177351662702175336, 2.80282201105271943883491328924, 3.86844686585152412526781788985, 4.17507071256542274008972909809, 4.85807083052575407576807696689, 5.42320271397944194786082854955, 5.93284244591264969411411532768, 5.94005669964667673908676027308, 6.79767342090842340018109565393, 7.43614633595531194995291118169, 7.892261528771830579960686916868, 8.005255546238632463989981397215, 8.493125287248278832640608416906, 9.059708945326420929361000907247, 9.681834509538176402191077040757, 9.770233097794993171001989060540, 10.40320328742561076757660148309, 10.72594872634854491426259276227

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