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Properties

Label 4-1184e2-1.1-c1e2-0-2
Degree $4$
Conductor $1401856$
Sign $1$
Analytic cond. $89.3835$
Root an. cond. $3.07478$
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 + 5-s − 2·7-s − 2·9-s + 5·11-s + 13-s − 15-s − 12·19-s + 2·21-s − 13·23-s − 6·25-s + 2·27-s + 17·29-s + 7·31-s − 5·33-s − 2·35-s − 2·37-s − 39-s − 3·41-s − 10·43-s − 2·45-s + 14·47-s + 2·49-s + 5·55-s + 12·57-s + 2·59-s + 23·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s − 2.75·19-s + 0.436·21-s − 2.71·23-s − 6/5·25-s + 0.384·27-s + 3.15·29-s + 1.25·31-s − 0.870·33-s − 0.338·35-s − 0.328·37-s − 0.160·39-s − 0.468·41-s − 1.52·43-s − 0.298·45-s + 2.04·47-s + 2/7·49-s + 0.674·55-s + 1.58·57-s + 0.260·59-s + 2.94·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1401856\)    =    \(2^{10} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(89.3835\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9425841540\)
\(L(\frac12)\) \(\approx\) \(0.9425841540\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_d
5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_h
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_c
11$D_{4}$ \( 1 - 5 T + 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_z
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_x
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$D_{4}$ \( 1 + 13 T + 85 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.23.n_dh
29$D_{4}$ \( 1 - 17 T + 127 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.29.ar_ex
31$D_{4}$ \( 1 - 7 T + 71 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.31.ah_ct
41$D_{4}$ \( 1 + 3 T + 81 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_dd
43$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_du
47$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_fa
53$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.53.a_cc
59$D_{4}$ \( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_ec
61$D_{4}$ \( 1 - 23 T + 251 T^{2} - 23 p T^{3} + p^{2} T^{4} \) 2.61.ax_jr
67$D_{4}$ \( 1 - 5 T + 137 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.67.af_fh
71$D_{4}$ \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_ec
73$D_{4}$ \( 1 + 5 T + 149 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.73.f_ft
79$D_{4}$ \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.79.n_hp
83$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.83.u_ig
89$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gg
97$D_{4}$ \( 1 - 16 T + 206 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_hy
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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

−10.09960582002453701257894410815, −9.963460755112151338624830905646, −8.943854945662757157937241373598, −8.674336089783039926021976792535, −8.463664478980694963502963240164, −8.216408754016141865913079152758, −7.42297529838422902198437690780, −6.72157624988627834397632366493, −6.48625207796035164603295262163, −6.30015135281531494270633288363, −5.86805185903757459088297366943, −5.63940767246395053833543933412, −4.55219772207833511488666846089, −4.45323139034592646033772308443, −3.88442803170269894227520402645, −3.49603303620635248158138670885, −2.47716742101795544253290290498, −2.30223276890911450164254648777, −1.46642681154165862007639293788, −0.43384436622137302076375624906, 0.43384436622137302076375624906, 1.46642681154165862007639293788, 2.30223276890911450164254648777, 2.47716742101795544253290290498, 3.49603303620635248158138670885, 3.88442803170269894227520402645, 4.45323139034592646033772308443, 4.55219772207833511488666846089, 5.63940767246395053833543933412, 5.86805185903757459088297366943, 6.30015135281531494270633288363, 6.48625207796035164603295262163, 6.72157624988627834397632366493, 7.42297529838422902198437690780, 8.216408754016141865913079152758, 8.463664478980694963502963240164, 8.674336089783039926021976792535, 8.943854945662757157937241373598, 9.963460755112151338624830905646, 10.09960582002453701257894410815

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