Properties

Label 4-490e2-1.1-c3e2-0-25
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 2·3-s + 12·4-s + 10·5-s + 8·6-s − 32·8-s − 33·9-s − 40·10-s − 26·11-s − 24·12-s + 102·13-s − 20·15-s + 80·16-s − 186·17-s + 132·18-s − 36·19-s + 120·20-s + 104·22-s − 44·23-s + 64·24-s + 75·25-s − 408·26-s + 86·27-s − 46·29-s + 80·30-s − 140·31-s − 192·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.384·3-s + 3/2·4-s + 0.894·5-s + 0.544·6-s − 1.41·8-s − 1.22·9-s − 1.26·10-s − 0.712·11-s − 0.577·12-s + 2.17·13-s − 0.344·15-s + 5/4·16-s − 2.65·17-s + 1.72·18-s − 0.434·19-s + 1.34·20-s + 1.00·22-s − 0.398·23-s + 0.544·24-s + 3/5·25-s − 3.07·26-s + 0.612·27-s − 0.294·29-s + 0.486·30-s − 0.811·31-s − 1.06·32-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 240100,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
5$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 37 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 26 T + 2703 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 102 T + 6833 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 186 T + 17897 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 36 T + 13530 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 44 T + 192 p T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 46 T + 38939 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 140 T + 38944 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 132 T + 28044 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 132 T + 100148 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 324 T + 181386 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 242 T + 215325 T^{2} + 242 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 208 T + 205512 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 780 T + 529576 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 216 T + 425298 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 256 T + 62452 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 692 T + 813066 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 832 T + 948202 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 1962 T + 1889271 T^{2} + 1962 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1712 T + 1530198 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1960 T + 2217986 T^{2} + 1960 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 102 T + 453465 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.27847497961099059308136024120, −10.01005595064845591322023479731, −9.144159992658927858809132564426, −8.972528334284393192014314925225, −8.539849586144574296658812875702, −8.537352906114624963476751133937, −7.64456748925158715219529819187, −7.22174967099541982022648114048, −6.41090971389846369568056675257, −6.33336961916090019116034444198, −5.67541778701140099340676040458, −5.65901961234132548134986104247, −4.46839766663131490311202300412, −4.04970000284833102842513607455, −2.80662065404681690344670275824, −2.76323900378937767482222063852, −1.79451695010414984815766840004, −1.33921706017389135457801376258, 0, 0, 1.33921706017389135457801376258, 1.79451695010414984815766840004, 2.76323900378937767482222063852, 2.80662065404681690344670275824, 4.04970000284833102842513607455, 4.46839766663131490311202300412, 5.65901961234132548134986104247, 5.67541778701140099340676040458, 6.33336961916090019116034444198, 6.41090971389846369568056675257, 7.22174967099541982022648114048, 7.64456748925158715219529819187, 8.537352906114624963476751133937, 8.539849586144574296658812875702, 8.972528334284393192014314925225, 9.144159992658927858809132564426, 10.01005595064845591322023479731, 10.27847497961099059308136024120

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