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Properties

Label 4-3024e2-1.1-c1e2-0-47
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  + 5·7-s − 9·13-s − 3·17-s + 19-s + 7·25-s + 9·29-s + 8·31-s + 11·37-s + 3·41-s + 15·43-s + 12·47-s + 18·49-s − 3·53-s − 12·59-s − 6·67-s − 21·73-s + 18·79-s + 9·83-s + 9·89-s − 45·91-s − 3·97-s + 14·103-s + 21·107-s − 5·109-s + 15·113-s − 15·119-s − 5·121-s + ⋯
L(s)  = 1  + 1.88·7-s − 2.49·13-s − 0.727·17-s + 0.229·19-s + 7/5·25-s + 1.67·29-s + 1.43·31-s + 1.80·37-s + 0.468·41-s + 2.28·43-s + 1.75·47-s + 18/7·49-s − 0.412·53-s − 1.56·59-s − 0.733·67-s − 2.45·73-s + 2.02·79-s + 0.987·83-s + 0.953·89-s − 4.71·91-s − 0.304·97-s + 1.37·103-s + 2.03·107-s − 0.478·109-s + 1.41·113-s − 1.37·119-s − 0.454·121-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

\(L(1)\) \(\approx\) \(3.787624378\)
\(L(\frac12)\) \(\approx\) \(3.787624378\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.j_bo
17$C_2^2$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_u
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \) 2.23.a_abr
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bh
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.al_dg
41$C_2^2$ \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_bs
43$C_2^2$ \( 1 - 15 T + 118 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_eo
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_dt
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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.61.a_cj
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.g_db
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2^2$ \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.73.v_im
79$C_2^2$ \( 1 - 18 T + 187 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.79.as_hf
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 - 9 T + 116 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.89.aj_em
97$C_2^2$ \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.97.d_dw
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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

−8.764777677888296816166031166533, −8.709017424301674372952073582497, −7.938607663679103672611090630952, −7.79236580921147253443676559403, −7.43806587835788716755593085354, −7.32027671286011565821919674474, −6.69055022127250126384016668605, −6.18542290816015122340750829361, −5.95353079476899549047780687450, −5.27370900823380735123599269235, −4.86703173852373272067892561903, −4.68594982752758964548158344186, −4.39864090089046212342919078238, −4.14718298362389576753381181831, −3.00001221276133776245267245096, −2.73678275423503414787766222719, −2.38370738715961523087949759289, −1.96205609745811646484540015950, −0.989867951004460234064315010659, −0.76289355498653388296075412773, 0.76289355498653388296075412773, 0.989867951004460234064315010659, 1.96205609745811646484540015950, 2.38370738715961523087949759289, 2.73678275423503414787766222719, 3.00001221276133776245267245096, 4.14718298362389576753381181831, 4.39864090089046212342919078238, 4.68594982752758964548158344186, 4.86703173852373272067892561903, 5.27370900823380735123599269235, 5.95353079476899549047780687450, 6.18542290816015122340750829361, 6.69055022127250126384016668605, 7.32027671286011565821919674474, 7.43806587835788716755593085354, 7.79236580921147253443676559403, 7.938607663679103672611090630952, 8.709017424301674372952073582497, 8.764777677888296816166031166533

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