Properties

Label 4-2850e2-1.1-c1e2-0-18
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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  + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s − 6·12-s + 6·13-s + 4·14-s + 5·16-s − 2·17-s + 6·18-s + 2·19-s − 4·21-s + 10·23-s − 8·24-s + 12·26-s − 4·27-s + 6·28-s − 8·29-s + 2·31-s + 6·32-s − 4·34-s + 9·36-s − 4·37-s + 4·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 1.73·12-s + 1.66·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.41·18-s + 0.458·19-s − 0.872·21-s + 2.08·23-s − 1.63·24-s + 2.35·26-s − 0.769·27-s + 1.13·28-s − 1.48·29-s + 0.359·31-s + 1.06·32-s − 0.685·34-s + 3/2·36-s − 0.657·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.872082328\)
\(L(\frac12)\) \(\approx\) \(6.872082328\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 20 T + 219 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 107 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 140 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 22 T + 255 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 111 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 155 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 72 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−8.983393638329352776948390823111, −8.638129522955471202031313618479, −7.88259923770659831535443333040, −7.86365645654584040463320761347, −7.16742568913924372614062482000, −6.99688805945246289905831484142, −6.48653844734527045112358021653, −6.30821987578752688608894062728, −5.63817197255964374087348118205, −5.57697346970628539897524518216, −4.98442629485407705728300081713, −4.97491738302514296442542307338, −4.19177010657332397664250594712, −4.02340333180061077855155384203, −3.49628878613053952275137333029, −3.10848679443823419206813481157, −2.34396472325476365757465147848, −1.88897408435967492974056121666, −1.17707155491389899811605176996, −0.824655969516385028097369408293, 0.824655969516385028097369408293, 1.17707155491389899811605176996, 1.88897408435967492974056121666, 2.34396472325476365757465147848, 3.10848679443823419206813481157, 3.49628878613053952275137333029, 4.02340333180061077855155384203, 4.19177010657332397664250594712, 4.97491738302514296442542307338, 4.98442629485407705728300081713, 5.57697346970628539897524518216, 5.63817197255964374087348118205, 6.30821987578752688608894062728, 6.48653844734527045112358021653, 6.99688805945246289905831484142, 7.16742568913924372614062482000, 7.86365645654584040463320761347, 7.88259923770659831535443333040, 8.638129522955471202031313618479, 8.983393638329352776948390823111

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