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Properties

Label 4-72e4-1.1-c1e2-0-3
Degree $4$
Conductor $26873856$
Sign $1$
Analytic cond. $1713.50$
Root an. cond. $6.43385$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1713.50\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26873856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.438297158\)
\(L(\frac12)\) \(\approx\) \(1.438297158\)
\(L(\frac{3}{2})\) 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.d_i
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.11.ac_x
13$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.13.f_y
17$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_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.h_co
37$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_by
41$D_{4}$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_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.ao_gl
61$D_{4}$ \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \) 2.61.ab_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.ah_gg
83$D_{4}$ \( 1 - 25 T + 314 T^{2} - 25 p T^{3} + p^{2} T^{4} \) 2.83.az_mc
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_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

−8.321283515254136534043356839556, −8.138629231776818740532688947296, −7.46624840873072606194571742725, −7.31347578766262663028993235959, −6.79010531190910557146903464271, −6.78772959649858258659454661174, −6.31132996824127025193929782001, −5.87388744257627378641659435332, −5.39104718504408689768664076557, −5.27119743145281330252013925459, −4.75408885331378748183654761453, −4.36531067291270112550403562757, −3.70762889340938545267862313589, −3.56362469534500090667721351235, −3.17599333462056849505940543695, −2.51315983834014644421953462582, −2.05473703962203636975436807099, −1.94523409503151156779672443085, −0.948106115100228775667493297987, −0.36310467537596153058026890106, 0.36310467537596153058026890106, 0.948106115100228775667493297987, 1.94523409503151156779672443085, 2.05473703962203636975436807099, 2.51315983834014644421953462582, 3.17599333462056849505940543695, 3.56362469534500090667721351235, 3.70762889340938545267862313589, 4.36531067291270112550403562757, 4.75408885331378748183654761453, 5.27119743145281330252013925459, 5.39104718504408689768664076557, 5.87388744257627378641659435332, 6.31132996824127025193929782001, 6.78772959649858258659454661174, 6.79010531190910557146903464271, 7.31347578766262663028993235959, 7.46624840873072606194571742725, 8.138629231776818740532688947296, 8.321283515254136534043356839556

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