Properties

Label 4-450e2-1.1-c5e2-0-21
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $5208.90$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 48·4-s + 100·7-s + 256·8-s − 540·11-s − 890·13-s + 800·14-s + 1.28e3·16-s − 492·17-s + 592·19-s − 4.32e3·22-s − 3.66e3·23-s − 7.12e3·26-s + 4.80e3·28-s − 5.70e3·29-s − 5.70e3·31-s + 6.14e3·32-s − 3.93e3·34-s − 1.13e4·37-s + 4.73e3·38-s − 1.54e4·41-s − 6.32e3·43-s − 2.59e4·44-s − 2.92e4·46-s + 7.80e3·47-s + 1.06e4·49-s − 4.27e4·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.771·7-s + 1.41·8-s − 1.34·11-s − 1.46·13-s + 1.09·14-s + 5/4·16-s − 0.412·17-s + 0.376·19-s − 1.90·22-s − 1.44·23-s − 2.06·26-s + 1.15·28-s − 1.25·29-s − 1.06·31-s + 1.06·32-s − 0.583·34-s − 1.35·37-s + 0.532·38-s − 1.43·41-s − 0.521·43-s − 2.01·44-s − 2.04·46-s + 0.515·47-s + 0.631·49-s − 2.19·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5208.90\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 202500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 100 T - 615 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 540 T + 248086 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 890 T + 793695 T^{2} + 890 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 492 T - 772670 T^{2} + 492 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 592 T + 4121589 T^{2} - 592 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 3660 T + 12548686 T^{2} + 3660 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 5700 T + 48997882 T^{2} + 5700 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 5708 T + 64485393 T^{2} + 5708 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 11300 T + 149454510 T^{2} + 11300 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 15420 T + 258100402 T^{2} + 15420 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 6320 T + 242261037 T^{2} + 6320 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 7800 T + 106610014 T^{2} - 7800 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 27828 T + 850018282 T^{2} - 27828 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 50520 T + 1968600982 T^{2} + 50520 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 29126 T + 1603768671 T^{2} + 29126 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 97400 T + 5071609653 T^{2} + 97400 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 6180 T + 3034897198 T^{2} + 6180 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 32900 T + 3887848086 T^{2} + 32900 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 7912 T + 1409684334 T^{2} - 7912 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 163464 T + 14190911110 T^{2} - 163464 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 164640 T + 13798144114 T^{2} + 164640 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 52430 T + 12995461155 T^{2} + 52430 p^{5} T^{3} + p^{10} T^{4} \)
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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

−10.14899779328195236069171196220, −9.935705106400373296396112955645, −9.061845747405042369247456006498, −8.734606535378019109346665991985, −7.80403298693254137593871834033, −7.75698230636487782360478728166, −7.28380768610713417135931686922, −6.85952698948006087644751169679, −6.02223665843315747869417568295, −5.67324472543686894398659633686, −5.09052714066856837845047209432, −4.95871698913845500016925936815, −4.29500523162034436698643985115, −3.75222339248455484122105984664, −3.07709350036095657466241134733, −2.54115146327264989585544592697, −1.88751517892199282527874122681, −1.61480369758448540050699361278, 0, 0, 1.61480369758448540050699361278, 1.88751517892199282527874122681, 2.54115146327264989585544592697, 3.07709350036095657466241134733, 3.75222339248455484122105984664, 4.29500523162034436698643985115, 4.95871698913845500016925936815, 5.09052714066856837845047209432, 5.67324472543686894398659633686, 6.02223665843315747869417568295, 6.85952698948006087644751169679, 7.28380768610713417135931686922, 7.75698230636487782360478728166, 7.80403298693254137593871834033, 8.734606535378019109346665991985, 9.061845747405042369247456006498, 9.935705106400373296396112955645, 10.14899779328195236069171196220

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