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Properties

Label 4-72e3-1.1-c1e2-0-22
Degree $4$
Conductor $373248$
Sign $-1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 9·23-s − 3·25-s − 5·29-s − 4·43-s + 13·47-s − 4·49-s − 17·53-s + 23·67-s − 3·71-s − 16·73-s − 4·97-s − 6·101-s − 9·115-s − 11·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 5·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.87·23-s − 3/5·25-s − 0.928·29-s − 0.609·43-s + 1.89·47-s − 4/7·49-s − 2.33·53-s + 2.80·67-s − 0.356·71-s − 1.87·73-s − 0.406·97-s − 0.597·101-s − 0.839·115-s − 121-s − 0.178·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.415·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(23.7986\)
Root analytic conductor: \(2.20870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.ab_e
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.13.a_am
17$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.17.a_ab
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.23.j_co
29$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.f_ck
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.37.a_ae
41$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.41.a_abl
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.e_cw
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.47.an_fg
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.r_gq
59$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.59.a_dh
61$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.61.a_dc
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) 2.67.ax_kg
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.d_dk
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.q_fq
79$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.79.a_afu
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.83.a_be
89$C_2^2$ \( 1 - 99 T^{2} + p^{2} T^{4} \) 2.89.a_adv
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.e_cc
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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

−8.285814911773320699378909658928, −8.128895191600339898630336615628, −7.54897503915688231652176416867, −7.14364621337242307941280571939, −6.48512198659618585116890288465, −6.07174699662810739821387495027, −5.72171370080198028338415339623, −5.16775822001417693146556910427, −4.57155919979003890081725580271, −3.91945789412065732999417151941, −3.59181037426724491969717023256, −2.68649758228303388344211089170, −2.09099002127923022917459852365, −1.44832722441958753262806348186, 0, 1.44832722441958753262806348186, 2.09099002127923022917459852365, 2.68649758228303388344211089170, 3.59181037426724491969717023256, 3.91945789412065732999417151941, 4.57155919979003890081725580271, 5.16775822001417693146556910427, 5.72171370080198028338415339623, 6.07174699662810739821387495027, 6.48512198659618585116890288465, 7.14364621337242307941280571939, 7.54897503915688231652176416867, 8.128895191600339898630336615628, 8.285814911773320699378909658928

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