Properties

Label 4-490e2-1.1-c3e2-0-8
Degree $4$
Conductor $240100$
Sign $1$
Analytic cond. $835.842$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 10·5-s + 50·9-s − 56·11-s + 16·16-s − 120·19-s − 40·20-s − 25·25-s − 180·29-s + 256·31-s − 200·36-s − 484·41-s + 224·44-s + 500·45-s − 560·55-s − 40·59-s − 1.08e3·61-s − 64·64-s − 2.25e3·71-s + 480·76-s + 1.44e3·79-s + 160·80-s + 1.77e3·81-s − 980·89-s − 1.20e3·95-s − 2.80e3·99-s + 100·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1.85·9-s − 1.53·11-s + 1/4·16-s − 1.44·19-s − 0.447·20-s − 1/5·25-s − 1.15·29-s + 1.48·31-s − 0.925·36-s − 1.84·41-s + 0.767·44-s + 1.65·45-s − 1.37·55-s − 0.0882·59-s − 2.27·61-s − 1/8·64-s − 3.77·71-s + 0.724·76-s + 2.05·79-s + 0.223·80-s + 2.42·81-s − 1.16·89-s − 1.29·95-s − 2.84·99-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(835.842\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 240100,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.621126635\)
\(L(\frac12)\) \(\approx\) \(1.621126635\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( 1 + p^{2} T^{2} \)
5$C_2$ \( 1 - 2 p T + p^{3} T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 50 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4250 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 5730 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 60 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 20970 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 90 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 128 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 45610 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 242 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 27970 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 156570 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 286090 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 542 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 413170 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 1128 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 378610 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 720 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 915090 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 490 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 294590 T^{2} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.59677049081958026467992305107, −10.28310959623063488724665626772, −10.04436436709044549703219310533, −9.336102705172220950322566356252, −9.309622555126909319751524018419, −8.338587278324295044867447015589, −8.250374680932885662051298296313, −7.47186657169440413599139992438, −7.29009788234279204657108503817, −6.51933442468180005342740096683, −6.19042967336081089797820315698, −5.57309162295232177041269287641, −5.04891229892299965853446587551, −4.45588919613803919402442398671, −4.28763849262282265775473743920, −3.36703067994750769363022655717, −2.69679450382151762891425293098, −1.83979462530784620400123479414, −1.59362054481535588627025835858, −0.39010393057023164507448278382, 0.39010393057023164507448278382, 1.59362054481535588627025835858, 1.83979462530784620400123479414, 2.69679450382151762891425293098, 3.36703067994750769363022655717, 4.28763849262282265775473743920, 4.45588919613803919402442398671, 5.04891229892299965853446587551, 5.57309162295232177041269287641, 6.19042967336081089797820315698, 6.51933442468180005342740096683, 7.29009788234279204657108503817, 7.47186657169440413599139992438, 8.250374680932885662051298296313, 8.338587278324295044867447015589, 9.309622555126909319751524018419, 9.336102705172220950322566356252, 10.04436436709044549703219310533, 10.28310959623063488724665626772, 10.59677049081958026467992305107

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