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Properties

Label 4-4788e2-1.1-c1e2-0-7
Degree $4$
Conductor $22924944$
Sign $1$
Analytic cond. $1461.71$
Root an. cond. $6.18323$
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  + 2·5-s − 2·7-s − 6·17-s + 2·19-s + 14·29-s − 12·31-s − 4·35-s − 4·41-s + 8·43-s + 14·47-s + 3·49-s + 22·53-s + 16·59-s + 8·61-s + 4·67-s + 2·71-s − 4·73-s + 4·79-s + 14·83-s − 12·85-s − 4·89-s + 4·95-s − 12·97-s + 30·101-s + 8·103-s + 22·107-s − 24·109-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 1.45·17-s + 0.458·19-s + 2.59·29-s − 2.15·31-s − 0.676·35-s − 0.624·41-s + 1.21·43-s + 2.04·47-s + 3/7·49-s + 3.02·53-s + 2.08·59-s + 1.02·61-s + 0.488·67-s + 0.237·71-s − 0.468·73-s + 0.450·79-s + 1.53·83-s − 1.30·85-s − 0.423·89-s + 0.410·95-s − 1.21·97-s + 2.98·101-s + 0.788·103-s + 2.12·107-s − 2.29·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22924944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22924944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22924944\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1461.71\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 22924944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.514105433\)
\(L(\frac12)\) \(\approx\) \(3.514105433\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_e
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 + 6 T + 36 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bk
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$D_{4}$ \( 1 - 14 T + 100 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.29.ao_dw
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.37.a_bu
41$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cg
43$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_cw
47$D_{4}$ \( 1 - 14 T + 136 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_fg
53$D_{4}$ \( 1 - 22 T + 220 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.53.aw_im
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_eg
67$D_{4}$ \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_eg
71$D_{4}$ \( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_dc
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$D_{4}$ \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_fe
83$D_{4}$ \( 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.83.ao_fw
89$D_{4}$ \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_fy
97$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_eo
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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

−8.508961923463231501674625371977, −8.342498511489194493128300538292, −7.55514568041951029706274017288, −7.27574074168221298503454521521, −7.00949586396019603865287813349, −6.70885970927139041592557610147, −6.24423250463898592017295345507, −5.99358146865372199238851536794, −5.46523503775127360361047023021, −5.35761649033743834776372244054, −4.83611213004660297604455525197, −4.29602087204456176418457931846, −3.91397625368970238504918286116, −3.66354092059792735001229663505, −3.00781247064544217646624377372, −2.45024797473173932349400087792, −2.29569875313131728817694321872, −1.87369371235495819777570242150, −0.822697844479424147022668890964, −0.68526133251869615826265961193, 0.68526133251869615826265961193, 0.822697844479424147022668890964, 1.87369371235495819777570242150, 2.29569875313131728817694321872, 2.45024797473173932349400087792, 3.00781247064544217646624377372, 3.66354092059792735001229663505, 3.91397625368970238504918286116, 4.29602087204456176418457931846, 4.83611213004660297604455525197, 5.35761649033743834776372244054, 5.46523503775127360361047023021, 5.99358146865372199238851536794, 6.24423250463898592017295345507, 6.70885970927139041592557610147, 7.00949586396019603865287813349, 7.27574074168221298503454521521, 7.55514568041951029706274017288, 8.342498511489194493128300538292, 8.508961923463231501674625371977

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