Properties

Label 4-2850e2-1.1-c1e2-0-21
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 $2$

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 − 4·7-s − 4·8-s + 3·9-s + 2·11-s − 6·12-s + 8·14-s + 5·16-s − 4·17-s − 6·18-s + 2·19-s + 8·21-s − 4·22-s + 2·23-s + 8·24-s − 4·27-s − 12·28-s + 6·29-s − 6·31-s − 6·32-s − 4·33-s + 8·34-s + 9·36-s − 4·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.51·7-s − 1.41·8-s + 9-s + 0.603·11-s − 1.73·12-s + 2.13·14-s + 5/4·16-s − 0.970·17-s − 1.41·18-s + 0.458·19-s + 1.74·21-s − 0.852·22-s + 0.417·23-s + 1.63·24-s − 0.769·27-s − 2.26·28-s + 1.11·29-s − 1.07·31-s − 1.06·32-s − 0.696·33-s + 1.37·34-s + 3/2·36-s − 0.657·37-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: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 10 T + 125 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 152 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 203 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 204 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.751451936220509576523778186765, −8.370096024881688421552663127788, −7.69779824562864070471344113973, −7.43433276418775594442952202263, −6.93421564401381579589844833502, −6.88607281996833567728995304571, −6.29606683765311064878301743121, −6.18672088483711867071126449190, −5.76122180477598959985503379021, −5.26627119432186950502629713842, −4.73631396413791374459885603786, −4.28122472671208880741835202545, −3.58456769113566184992346954775, −3.43958132187156480106481413045, −2.52987735606926522428771591853, −2.43629367969943340209023668267, −1.29944804175303045111882703694, −1.19149362302376323593201201087, 0, 0, 1.19149362302376323593201201087, 1.29944804175303045111882703694, 2.43629367969943340209023668267, 2.52987735606926522428771591853, 3.43958132187156480106481413045, 3.58456769113566184992346954775, 4.28122472671208880741835202545, 4.73631396413791374459885603786, 5.26627119432186950502629713842, 5.76122180477598959985503379021, 6.18672088483711867071126449190, 6.29606683765311064878301743121, 6.88607281996833567728995304571, 6.93421564401381579589844833502, 7.43433276418775594442952202263, 7.69779824562864070471344113973, 8.370096024881688421552663127788, 8.751451936220509576523778186765

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