Properties

Label 4-9280e2-1.1-c1e2-0-3
Degree $4$
Conductor $86118400$
Sign $1$
Analytic cond. $5490.98$
Root an. cond. $8.60820$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 3·7-s − 2·9-s − 2·11-s + 13-s + 2·15-s − 17-s − 6·19-s − 3·21-s − 3·23-s + 3·25-s + 2·27-s + 2·29-s + 17·31-s + 2·33-s − 6·35-s − 6·37-s − 39-s + 10·41-s − 3·43-s + 4·45-s − 4·47-s − 4·49-s + 51-s + 23·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s − 0.242·17-s − 1.37·19-s − 0.654·21-s − 0.625·23-s + 3/5·25-s + 0.384·27-s + 0.371·29-s + 3.05·31-s + 0.348·33-s − 1.01·35-s − 0.986·37-s − 0.160·39-s + 1.56·41-s − 0.457·43-s + 0.596·45-s − 0.583·47-s − 4/7·49-s + 0.140·51-s + 3.15·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86118400\)    =    \(2^{12} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5490.98\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86118400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934657046\)
\(L(\frac12)\) \(\approx\) \(1.934657046\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 17 T + 131 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 85 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 23 T + 235 T^{2} - 23 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 19 T + 205 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 7 T + 141 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 47 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−8.040238746342811027851328209496, −7.71743874008209662530113248830, −7.00137798718861777984936299869, −6.92117379895445576694634026183, −6.57128353917841588338626012692, −6.21378254838669061667810924811, −5.70635770787159541235960505990, −5.54022410152369497389714665806, −5.00263037829826218848876993489, −4.90537672193263815432610030428, −4.28418282384966645675414096730, −4.18672007641034598332804970205, −3.81566968504795696422827682831, −3.27865110585940698022067001885, −2.63026551443280576209212871642, −2.48985683947413963307001891060, −2.08055657171538410739285706832, −1.38758611397023972147259078888, −0.71994881909178174157757544823, −0.48343060225222251969999897540, 0.48343060225222251969999897540, 0.71994881909178174157757544823, 1.38758611397023972147259078888, 2.08055657171538410739285706832, 2.48985683947413963307001891060, 2.63026551443280576209212871642, 3.27865110585940698022067001885, 3.81566968504795696422827682831, 4.18672007641034598332804970205, 4.28418282384966645675414096730, 4.90537672193263815432610030428, 5.00263037829826218848876993489, 5.54022410152369497389714665806, 5.70635770787159541235960505990, 6.21378254838669061667810924811, 6.57128353917841588338626012692, 6.92117379895445576694634026183, 7.00137798718861777984936299869, 7.71743874008209662530113248830, 8.040238746342811027851328209496

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