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Properties

Label 4-560e2-1.1-c1e2-0-28
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
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  + 3-s + 2·5-s + 2·7-s − 9-s − 11-s + 5·13-s + 2·15-s − 5·17-s + 6·19-s + 2·21-s + 2·23-s + 3·25-s + 29-s − 33-s + 4·35-s + 12·37-s + 5·39-s + 2·41-s − 10·43-s − 2·45-s + 5·47-s + 3·49-s − 5·51-s − 2·53-s − 2·55-s + 6·57-s + 8·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.755·7-s − 1/3·9-s − 0.301·11-s + 1.38·13-s + 0.516·15-s − 1.21·17-s + 1.37·19-s + 0.436·21-s + 0.417·23-s + 3/5·25-s + 0.185·29-s − 0.174·33-s + 0.676·35-s + 1.97·37-s + 0.800·39-s + 0.312·41-s − 1.52·43-s − 0.298·45-s + 0.729·47-s + 3/7·49-s − 0.700·51-s − 0.274·53-s − 0.269·55-s + 0.794·57-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.082828583\)
\(L(\frac12)\) \(\approx\) \(3.082828583\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_s
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_bc
17$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_bk
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_be
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_be
29$D_{4}$ \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_u
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_co
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dq
47$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_ck
53$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_dm
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$D_{4}$ \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_aw
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_cs
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_dq
79$D_{4}$ \( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_gs
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$D_{4}$ \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_go
97$D_{4}$ \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_ee
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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

−10.84887930448418842093346746469, −10.71852278879301063064640206695, −9.996872262174677903093072110117, −9.665051256331339298262348336268, −9.108919882625151785185346284674, −8.827835147304446976532813852039, −8.351613035344276371663628470056, −8.092656691658838401609900676659, −7.40369311082293681903432731110, −7.00222461019926043667084908705, −6.31541053035408999850880288606, −6.02299401560137118713179867984, −5.37801526421910066501971091992, −5.04595920059395037042341789054, −4.30353083834975357581212378203, −3.84123208348491185714193855596, −2.78921232307861910493038679196, −2.78687918473758351130938627062, −1.74497648316645092426880643265, −1.10076043135417396284922284686, 1.10076043135417396284922284686, 1.74497648316645092426880643265, 2.78687918473758351130938627062, 2.78921232307861910493038679196, 3.84123208348491185714193855596, 4.30353083834975357581212378203, 5.04595920059395037042341789054, 5.37801526421910066501971091992, 6.02299401560137118713179867984, 6.31541053035408999850880288606, 7.00222461019926043667084908705, 7.40369311082293681903432731110, 8.092656691658838401609900676659, 8.351613035344276371663628470056, 8.827835147304446976532813852039, 9.108919882625151785185346284674, 9.665051256331339298262348336268, 9.996872262174677903093072110117, 10.71852278879301063064640206695, 10.84887930448418842093346746469

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