Properties

Label 4-405e2-1.1-c3e2-0-2
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $571.007$
Root an. cond. $4.88833$
Motivic weight $3$
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  + 5·2-s + 8·4-s − 5·5-s − 9·7-s − 5·8-s − 25·10-s + 8·11-s − 43·13-s − 45·14-s − 25·16-s − 244·17-s − 118·19-s − 40·20-s + 40·22-s + 213·23-s − 215·26-s − 72·28-s − 224·29-s + 36·31-s − 40·32-s − 1.22e3·34-s + 45·35-s + 412·37-s − 590·38-s + 25·40-s − 413·41-s + 392·43-s + ⋯
L(s)  = 1  + 1.76·2-s + 4-s − 0.447·5-s − 0.485·7-s − 0.220·8-s − 0.790·10-s + 0.219·11-s − 0.917·13-s − 0.859·14-s − 0.390·16-s − 3.48·17-s − 1.42·19-s − 0.447·20-s + 0.387·22-s + 1.93·23-s − 1.62·26-s − 0.485·28-s − 1.43·29-s + 0.208·31-s − 0.220·32-s − 6.15·34-s + 0.217·35-s + 1.83·37-s − 2.51·38-s + 0.0988·40-s − 1.57·41-s + 1.39·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(571.007\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 164025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6161183210\)
\(L(\frac12)\) \(\approx\) \(0.6161183210\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5$C_2$ \( 1 + p T + p^{2} T^{2} \)
good2$C_2^2$ \( 1 - 5 T + 17 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 9 T - 262 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 8 T - 1267 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 43 T - 348 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 122 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 59 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 213 T + 33202 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 224 T + 25787 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 36 T - 28495 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 206 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 413 T + 101648 T^{2} + 413 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 392 T + 74157 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 311 T - 7102 T^{2} - 311 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 377 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 337 T - 91810 T^{2} + 337 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 40 T - 225381 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 348 T - 179659 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 62 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 1214 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 294 T - 406603 T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 534 T - 286631 T^{2} + 534 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 810 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 928 T - 51489 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.69800744582416750952454736064, −10.75026124004093082385894414401, −10.55319215565430941041496125229, −9.387259108296648797379309412181, −9.345865679972970398183574779371, −8.794063490711800750748397891028, −8.474743232255682079322211580297, −7.47756531622510158647089762337, −7.20488945951270260611466606195, −6.65107541746026602479410518852, −6.21123123898673196640124080529, −5.74571551970640828631512640304, −4.95506520429868158166634665382, −4.48741198414787853081740684821, −4.33752899852849617482190129252, −3.90145598083978401579970537865, −2.87662664633334554027952456553, −2.63967504927610548993833761275, −1.72949167696162632707072665513, −0.18418789032562324421490103328, 0.18418789032562324421490103328, 1.72949167696162632707072665513, 2.63967504927610548993833761275, 2.87662664633334554027952456553, 3.90145598083978401579970537865, 4.33752899852849617482190129252, 4.48741198414787853081740684821, 4.95506520429868158166634665382, 5.74571551970640828631512640304, 6.21123123898673196640124080529, 6.65107541746026602479410518852, 7.20488945951270260611466606195, 7.47756531622510158647089762337, 8.474743232255682079322211580297, 8.794063490711800750748397891028, 9.345865679972970398183574779371, 9.387259108296648797379309412181, 10.55319215565430941041496125229, 10.75026124004093082385894414401, 11.69800744582416750952454736064

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