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Properties

Label 4-2268e2-1.1-c1e2-0-10
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $327.974$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 7-s − 9·11-s − 6·13-s + 12·17-s − 9·23-s + 5·25-s + 18·29-s − 9·31-s + 3·35-s + 2·37-s − 3·41-s − 10·43-s − 6·47-s − 6·49-s − 27·55-s + 6·59-s − 24·61-s − 18·65-s − 2·67-s − 9·77-s − 14·79-s + 6·83-s + 36·85-s − 18·89-s − 6·91-s − 12·97-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 2.71·11-s − 1.66·13-s + 2.91·17-s − 1.87·23-s + 25-s + 3.34·29-s − 1.61·31-s + 0.507·35-s + 0.328·37-s − 0.468·41-s − 1.52·43-s − 0.875·47-s − 6/7·49-s − 3.64·55-s + 0.781·59-s − 3.07·61-s − 2.23·65-s − 0.244·67-s − 1.02·77-s − 1.57·79-s + 0.658·83-s + 3.90·85-s − 1.90·89-s − 0.628·91-s − 1.21·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(327.974\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542462125\)
\(L(\frac12)\) \(\approx\) \(1.542462125\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.11.j_bm
13$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_z
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_by
29$C_2^2$ \( 1 - 18 T + 137 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.29.as_fh
31$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.31.j_cg
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2^2$ \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.61.y_jt
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \) 2.71.a_ael
73$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.73.a_afe
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_abv
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2^2$ \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_fp
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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

−9.528603442246488185769251734189, −8.564139481254832215585719344567, −8.477890694291114219936246941255, −7.85377480890382460872734237911, −7.83183766628998824800506525076, −7.46023629192446959198221374310, −7.00423361669982307031468396996, −6.24133944816750565531342282944, −6.10985067621884162647043114856, −5.52857665224749645380087159067, −5.25267778488987901645138101244, −4.92960570054791929779174645553, −4.79567275951271402140185945471, −3.95213466285539173409362591479, −3.06282524784053394291760056659, −2.88768169706976723533806931641, −2.65329439272408476644445287564, −1.74010894474790852585693735195, −1.60805998744677780583108325499, −0.40003101924768251832120590587, 0.40003101924768251832120590587, 1.60805998744677780583108325499, 1.74010894474790852585693735195, 2.65329439272408476644445287564, 2.88768169706976723533806931641, 3.06282524784053394291760056659, 3.95213466285539173409362591479, 4.79567275951271402140185945471, 4.92960570054791929779174645553, 5.25267778488987901645138101244, 5.52857665224749645380087159067, 6.10985067621884162647043114856, 6.24133944816750565531342282944, 7.00423361669982307031468396996, 7.46023629192446959198221374310, 7.83183766628998824800506525076, 7.85377480890382460872734237911, 8.477890694291114219936246941255, 8.564139481254832215585719344567, 9.528603442246488185769251734189

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