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Properties

Label 4-3024e2-1.1-c1e2-0-23
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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  + 6·5-s + 7-s − 6·11-s − 5·13-s + 3·17-s + 5·19-s − 6·23-s + 17·25-s − 3·29-s − 4·31-s + 6·35-s + 7·37-s − 9·41-s + 11·43-s − 6·49-s − 3·53-s − 36·55-s − 12·59-s − 2·61-s − 30·65-s − 4·67-s − 11·73-s − 6·77-s + 8·79-s − 3·83-s + 18·85-s + 15·89-s + ⋯
L(s)  = 1  + 2.68·5-s + 0.377·7-s − 1.80·11-s − 1.38·13-s + 0.727·17-s + 1.14·19-s − 1.25·23-s + 17/5·25-s − 0.557·29-s − 0.718·31-s + 1.01·35-s + 1.15·37-s − 1.40·41-s + 1.67·43-s − 6/7·49-s − 0.412·53-s − 4.85·55-s − 1.56·59-s − 0.256·61-s − 3.72·65-s − 0.488·67-s − 1.28·73-s − 0.683·77-s + 0.900·79-s − 0.329·83-s + 1.95·85-s + 1.58·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.189497559\)
\(L(\frac12)\) \(\approx\) \(3.189497559\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.43.al_da
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ap
83$C_2^2$ \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_acw
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ads
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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

−9.211469072734228254881094324105, −8.540529748498782536428073311394, −8.122894446217551394961610084594, −7.59759424138145145550802511522, −7.48607038340698147979430867781, −7.26722923731764579120596515264, −6.40065857029883971006959080547, −6.02872741454883456473477649065, −5.93744544711787277488291436519, −5.45475254456205444290706415813, −5.19870034944944628669699954424, −4.84644616104127240115935144689, −4.52546400025608236812179144421, −3.64222240930301489433243536970, −3.05401588831580605209632507910, −2.69108871259168185271216552364, −2.29445083035991655851967402962, −1.79509243681495243575394538969, −1.58430333699523206065321024565, −0.51093835662034441770303603523, 0.51093835662034441770303603523, 1.58430333699523206065321024565, 1.79509243681495243575394538969, 2.29445083035991655851967402962, 2.69108871259168185271216552364, 3.05401588831580605209632507910, 3.64222240930301489433243536970, 4.52546400025608236812179144421, 4.84644616104127240115935144689, 5.19870034944944628669699954424, 5.45475254456205444290706415813, 5.93744544711787277488291436519, 6.02872741454883456473477649065, 6.40065857029883971006959080547, 7.26722923731764579120596515264, 7.48607038340698147979430867781, 7.59759424138145145550802511522, 8.122894446217551394961610084594, 8.540529748498782536428073311394, 9.211469072734228254881094324105

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