Properties

Label 4-3840e2-1.1-c1e2-0-19
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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  − 9-s + 12·17-s − 16·23-s − 25-s + 8·31-s − 4·41-s + 16·47-s − 14·49-s − 16·71-s + 28·73-s − 24·79-s + 81-s + 28·89-s + 4·97-s − 16·103-s − 20·113-s + 22·121-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  − 1/3·9-s + 2.91·17-s − 3.33·23-s − 1/5·25-s + 1.43·31-s − 0.624·41-s + 2.33·47-s − 2·49-s − 1.89·71-s + 3.27·73-s − 2.70·79-s + 1/9·81-s + 2.96·89-s + 0.406·97-s − 1.57·103-s − 1.88·113-s + 2·121-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800064539\)
\(L(\frac12)\) \(\approx\) \(1.800064539\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.650999426864013446260208181150, −8.035570985380482134476749926426, −7.965743589608975679233923328142, −7.71604590658453268096394262956, −7.47041099164964697221049263451, −6.70192636153825363612406782316, −6.37962823724393061817821724156, −6.02953872046730374709736573289, −5.78690743690175185560878556349, −5.30441820394794900917805333909, −5.08152324919443367659255227682, −4.40475011416131343569874803981, −4.02502835484179707680358310849, −3.55120584020870056508697459347, −3.41670308392441788508564955804, −2.55338153179805031541196666305, −2.46609638879247422239295151276, −1.56169011630807567410406283985, −1.26802235711140916350607979965, −0.40653967383218646315047504887, 0.40653967383218646315047504887, 1.26802235711140916350607979965, 1.56169011630807567410406283985, 2.46609638879247422239295151276, 2.55338153179805031541196666305, 3.41670308392441788508564955804, 3.55120584020870056508697459347, 4.02502835484179707680358310849, 4.40475011416131343569874803981, 5.08152324919443367659255227682, 5.30441820394794900917805333909, 5.78690743690175185560878556349, 6.02953872046730374709736573289, 6.37962823724393061817821724156, 6.70192636153825363612406782316, 7.47041099164964697221049263451, 7.71604590658453268096394262956, 7.965743589608975679233923328142, 8.035570985380482134476749926426, 8.650999426864013446260208181150

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