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Properties

Label 4-4725e2-1.1-c1e2-0-12
Degree $4$
Conductor $22325625$
Sign $1$
Analytic cond. $1423.50$
Root an. cond. $6.14241$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·4-s + 2·7-s − 3·8-s + 4·11-s − 5·13-s + 2·14-s + 16-s − 17-s − 3·19-s + 4·22-s − 13·23-s − 5·26-s − 4·28-s + 29-s + 2·32-s − 34-s − 3·38-s + 5·41-s + 2·43-s − 8·44-s − 13·46-s − 22·47-s + 3·49-s + 10·52-s + 11·53-s − 6·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 0.755·7-s − 1.06·8-s + 1.20·11-s − 1.38·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 0.852·22-s − 2.71·23-s − 0.980·26-s − 0.755·28-s + 0.185·29-s + 0.353·32-s − 0.171·34-s − 0.486·38-s + 0.780·41-s + 0.304·43-s − 1.20·44-s − 1.91·46-s − 3.20·47-s + 3/7·49-s + 1.38·52-s + 1.51·53-s − 0.801·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22325625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22325625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22325625\)    =    \(3^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1423.50\)
Root analytic conductor: \(6.14241\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 22325625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_v
13$D_{4}$ \( 1 + 5 T + 31 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.13.f_bf
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_bh
19$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_j
23$D_{4}$ \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.23.n_dj
29$D_{4}$ \( 1 - T - 43 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_abr
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.31.a_r
37$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.37.a_bd
41$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.41.af_cf
43$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_h
47$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.47.w_ih
53$D_{4}$ \( 1 - 11 T + 135 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.53.al_ff
59$D_{4}$ \( 1 - 16 T + 137 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.59.aq_fh
61$D_{4}$ \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.61.v_in
67$D_{4}$ \( 1 + 7 T + 85 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_dh
71$D_{4}$ \( 1 - T + 131 T^{2} - p T^{3} + p^{2} T^{4} \) 2.71.ab_fb
73$D_{4}$ \( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_dn
79$D_{4}$ \( 1 - 9 T + 177 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_gv
83$D_{4}$ \( 1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_fh
89$D_{4}$ \( 1 + 6 T + 167 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_gl
97$D_{4}$ \( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_fx
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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

−8.112065326360368691139937715618, −7.86076385646026684808938831686, −7.42371253359584551199346309876, −6.98511578389513866648498642243, −6.61523821622143271777947886324, −6.17061956067114644986514432445, −5.85139157010994271072192767213, −5.53997603013213380483032796126, −4.98563643187102024486243805788, −4.69917038922586162819833893291, −4.27943106467769007981420215438, −4.25791517684523557601567061181, −3.67233238001055450591680162641, −3.39647652184184603421936063801, −2.54959027754455251668857274972, −2.28212718239631039229095044207, −1.68811323508073191986973099598, −1.20776691220666590698986010322, 0, 0, 1.20776691220666590698986010322, 1.68811323508073191986973099598, 2.28212718239631039229095044207, 2.54959027754455251668857274972, 3.39647652184184603421936063801, 3.67233238001055450591680162641, 4.25791517684523557601567061181, 4.27943106467769007981420215438, 4.69917038922586162819833893291, 4.98563643187102024486243805788, 5.53997603013213380483032796126, 5.85139157010994271072192767213, 6.17061956067114644986514432445, 6.61523821622143271777947886324, 6.98511578389513866648498642243, 7.42371253359584551199346309876, 7.86076385646026684808938831686, 8.112065326360368691139937715618

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