Properties

Label 4-272e2-1.1-c1e2-0-6
Degree $4$
Conductor $73984$
Sign $1$
Analytic cond. $4.71728$
Root an. cond. $1.47374$
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  − 2·9-s + 4·13-s − 6·17-s + 8·19-s + 2·25-s + 8·43-s + 14·49-s + 12·53-s − 24·59-s + 8·67-s − 5·81-s + 24·83-s + 12·89-s − 12·101-s − 16·103-s − 8·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2/3·9-s + 1.10·13-s − 1.45·17-s + 1.83·19-s + 2/5·25-s + 1.21·43-s + 2·49-s + 1.64·53-s − 3.12·59-s + 0.977·67-s − 5/9·81-s + 2.63·83-s + 1.27·89-s − 1.19·101-s − 1.57·103-s − 0.739·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(73984\)    =    \(2^{8} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(4.71728\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 73984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435415367\)
\(L(\frac12)\) \(\approx\) \(1.435415367\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−12.05257965953215308292165099761, −11.73305834041596980820387698507, −11.06560733655753389956397479409, −10.82300603427881094733017023942, −10.50140988361708974737737921690, −9.554735240840037024238594334444, −9.321252973890626987575460728023, −8.757486953765343048268981151521, −8.485191488614629887960143244156, −7.67534224193134361128247481582, −7.35653237252661275065747417873, −6.66617054273959782992272002749, −6.11563698043049732751454434838, −5.64451298682486971010688452189, −5.05998546438718223681130878295, −4.32852504418706220349484280857, −3.68464741475710038172492983459, −2.98820601994076078256744397564, −2.23914169154826337672882037980, −0.994569142519157220536792371076, 0.994569142519157220536792371076, 2.23914169154826337672882037980, 2.98820601994076078256744397564, 3.68464741475710038172492983459, 4.32852504418706220349484280857, 5.05998546438718223681130878295, 5.64451298682486971010688452189, 6.11563698043049732751454434838, 6.66617054273959782992272002749, 7.35653237252661275065747417873, 7.67534224193134361128247481582, 8.485191488614629887960143244156, 8.757486953765343048268981151521, 9.321252973890626987575460728023, 9.554735240840037024238594334444, 10.50140988361708974737737921690, 10.82300603427881094733017023942, 11.06560733655753389956397479409, 11.73305834041596980820387698507, 12.05257965953215308292165099761

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