Properties

Label 4-9280e2-1.1-c1e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 5·7-s − 2·9-s + 2·11-s − 9·13-s − 2·15-s + 3·17-s − 6·19-s − 5·21-s + 7·23-s + 3·25-s + 2·27-s − 2·29-s − 5·31-s − 2·33-s + 10·35-s + 2·37-s + 9·39-s − 10·41-s − 3·43-s − 4·45-s + 8·49-s − 3·51-s − 3·53-s + 4·55-s + 6·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.88·7-s − 2/3·9-s + 0.603·11-s − 2.49·13-s − 0.516·15-s + 0.727·17-s − 1.37·19-s − 1.09·21-s + 1.45·23-s + 3/5·25-s + 0.384·27-s − 0.371·29-s − 0.898·31-s − 0.348·33-s + 1.69·35-s + 0.328·37-s + 1.44·39-s − 1.56·41-s − 0.457·43-s − 0.596·45-s + 8/7·49-s − 0.420·51-s − 0.412·53-s + 0.539·55-s + 0.794·57-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

\(L(1)\) \(\approx\) \(1.657990469\)
\(L(\frac12)\) \(\approx\) \(1.657990469\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 55 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 109 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 17 T + 165 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 13 T + 107 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 129 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 22 T + 286 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 13 T + 207 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−7.81232037174815860388864132363, −7.40400676578107535183481223185, −7.24452251628153117929420333711, −6.99253226687698549816358023514, −6.35501545023042355922775399123, −6.27039632183739808554896500550, −5.68859506132883085228963775518, −5.35259388479413981353570790705, −5.18555450510944312644244373233, −4.80642359458410763650274264514, −4.64191513305701036418463827657, −4.31768051937798979890076360693, −3.46243441087194718245863116851, −3.35852217895248572593957720382, −2.54029732528792375600464846164, −2.42705383394039339561400295700, −1.77301387109249506588182094316, −1.73578263687636999331627064665, −1.02710763682646817946208950244, −0.31016099561038437563434885158, 0.31016099561038437563434885158, 1.02710763682646817946208950244, 1.73578263687636999331627064665, 1.77301387109249506588182094316, 2.42705383394039339561400295700, 2.54029732528792375600464846164, 3.35852217895248572593957720382, 3.46243441087194718245863116851, 4.31768051937798979890076360693, 4.64191513305701036418463827657, 4.80642359458410763650274264514, 5.18555450510944312644244373233, 5.35259388479413981353570790705, 5.68859506132883085228963775518, 6.27039632183739808554896500550, 6.35501545023042355922775399123, 6.99253226687698549816358023514, 7.24452251628153117929420333711, 7.40400676578107535183481223185, 7.81232037174815860388864132363

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