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Properties

Label 4-4752e2-1.1-c1e2-0-13
Degree $4$
Conductor $22581504$
Sign $1$
Analytic cond. $1439.81$
Root an. cond. $6.15994$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s − 4·17-s + 6·23-s − 10·25-s − 8·29-s − 8·31-s − 6·37-s − 4·41-s − 4·43-s + 2·47-s − 7·49-s − 8·53-s + 14·59-s − 16·67-s + 8·71-s − 8·73-s + 8·79-s + 16·83-s − 16·89-s − 6·97-s − 16·101-s − 8·103-s − 4·109-s − 16·113-s + 3·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.603·11-s − 0.970·17-s + 1.25·23-s − 2·25-s − 1.48·29-s − 1.43·31-s − 0.986·37-s − 0.624·41-s − 0.609·43-s + 0.291·47-s − 49-s − 1.09·53-s + 1.82·59-s − 1.95·67-s + 0.949·71-s − 0.936·73-s + 0.900·79-s + 1.75·83-s − 1.69·89-s − 0.609·97-s − 1.59·101-s − 0.788·103-s − 0.383·109-s − 1.50·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22581504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22581504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22581504\)    =    \(2^{8} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1439.81\)
Root analytic conductor: \(6.15994\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 22581504,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.7.a_h
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 + 4 T + 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_bf
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$D_{4}$ \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bb
29$D_{4}$ \( 1 + 8 T + 67 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cp
31$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_by
37$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_cd
41$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_x
43$D_{4}$ \( 1 + 4 T + 83 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_df
47$D_{4}$ \( 1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_cp
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.59.ao_gl
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.61.a_k
67$D_{4}$ \( 1 + 16 T + 170 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.67.q_go
71$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_bu
73$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_by
79$D_{4}$ \( 1 - 8 T + 111 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_eh
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$D_{4}$ \( 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ig
97$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_dn
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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

−8.010538207577626195840468312322, −7.83176239421351398994892990010, −7.25852169707008241412059015191, −7.11866061571100045285641904523, −6.56644134774103095424577332083, −6.51734332761957777313561505085, −5.78553453893309239180412491488, −5.64672998102455992241222380044, −5.02495152364437868334330843322, −5.02165417874334781031430941157, −4.12552475907476776713740201403, −4.07565257412990917913935909007, −3.50881751156537294124578549875, −3.31228705090494769298892673910, −2.54875870170649145034128017817, −2.13702423410529552117907671606, −1.60777098438092184837572495451, −1.32393309398928847411831885048, 0, 0, 1.32393309398928847411831885048, 1.60777098438092184837572495451, 2.13702423410529552117907671606, 2.54875870170649145034128017817, 3.31228705090494769298892673910, 3.50881751156537294124578549875, 4.07565257412990917913935909007, 4.12552475907476776713740201403, 5.02165417874334781031430941157, 5.02495152364437868334330843322, 5.64672998102455992241222380044, 5.78553453893309239180412491488, 6.51734332761957777313561505085, 6.56644134774103095424577332083, 7.11866061571100045285641904523, 7.25852169707008241412059015191, 7.83176239421351398994892990010, 8.010538207577626195840468312322

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