Properties

Label 4-9800e2-1.1-c1e2-0-8
Degree $4$
Conductor $96040000$
Sign $1$
Analytic cond. $6123.59$
Root an. cond. $8.84609$
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  + 2·3-s − 9-s − 4·11-s − 4·13-s + 4·17-s + 14·23-s − 6·27-s − 10·29-s − 4·31-s − 8·33-s − 8·39-s + 6·41-s − 6·43-s − 12·47-s + 8·51-s − 8·59-s + 2·61-s + 6·67-s + 28·69-s − 24·71-s + 4·73-s − 8·79-s − 4·81-s + 18·83-s − 20·87-s − 22·89-s − 8·93-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.970·17-s + 2.91·23-s − 1.15·27-s − 1.85·29-s − 0.718·31-s − 1.39·33-s − 1.28·39-s + 0.937·41-s − 0.914·43-s − 1.75·47-s + 1.12·51-s − 1.04·59-s + 0.256·61-s + 0.733·67-s + 3.37·69-s − 2.84·71-s + 0.468·73-s − 0.900·79-s − 4/9·81-s + 1.97·83-s − 2.14·87-s − 2.33·89-s − 0.829·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(96040000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(6123.59\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 96040000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 14 T + 93 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T - 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 2 T + 91 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−7.36704910371947464701627265874, −7.31190273626626586122409322862, −7.07526705545864904323204606151, −6.67355071855595058923641126825, −5.91118536937455914598724625034, −5.78023028878370388026184991397, −5.45970871883817431002797705811, −5.07798928232015481359769948541, −4.64586212396625846765261830298, −4.63910898773724583214304905114, −3.72359487845364375365054891267, −3.46894692130880219024768769752, −3.05556635382773581164498866073, −2.98188273920396512757567966598, −2.41389015489771500166816105683, −2.22158897405785341119485150975, −1.48617882024117632317609111240, −1.13761349690508980669908107473, 0, 0, 1.13761349690508980669908107473, 1.48617882024117632317609111240, 2.22158897405785341119485150975, 2.41389015489771500166816105683, 2.98188273920396512757567966598, 3.05556635382773581164498866073, 3.46894692130880219024768769752, 3.72359487845364375365054891267, 4.63910898773724583214304905114, 4.64586212396625846765261830298, 5.07798928232015481359769948541, 5.45970871883817431002797705811, 5.78023028878370388026184991397, 5.91118536937455914598724625034, 6.67355071855595058923641126825, 7.07526705545864904323204606151, 7.31190273626626586122409322862, 7.36704910371947464701627265874

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