Properties

Label 4-1815e2-1.1-c3e2-0-0
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $11467.9$
Root an. cond. $10.3483$
Motivic weight $3$
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  + 6·3-s − 4·4-s + 10·5-s + 27·9-s − 24·12-s + 60·15-s − 48·16-s − 40·20-s + 390·23-s + 75·25-s + 108·27-s + 194·31-s − 108·36-s + 548·37-s + 270·45-s + 6·47-s − 288·48-s + 286·49-s + 6·53-s − 1.29e3·59-s − 240·60-s + 448·64-s + 140·67-s + 2.34e3·69-s − 744·71-s + 450·75-s − 480·80-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 0.894·5-s + 9-s − 0.577·12-s + 1.03·15-s − 3/4·16-s − 0.447·20-s + 3.53·23-s + 3/5·25-s + 0.769·27-s + 1.12·31-s − 1/2·36-s + 2.43·37-s + 0.894·45-s + 0.0186·47-s − 0.866·48-s + 0.833·49-s + 0.0155·53-s − 2.85·59-s − 0.516·60-s + 7/8·64-s + 0.255·67-s + 4.08·69-s − 1.24·71-s + 0.692·75-s − 0.670·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(11467.9\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.093823550\)
\(L(\frac12)\) \(\approx\) \(8.093823550\)
\(L(\frac{5}{2})\) 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)$
bad3$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 - p T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + p^{2} T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 286 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 1414 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 7639 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 12746 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 195 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 47578 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 97 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 274 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 10010 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 13786 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 3 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + 648 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 243287 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 372 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 96766 T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 714275 T^{2} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 1141546 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1122 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 470 T + p^{3} T^{2} )^{2} \)
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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

−9.020415492340436091207147850099, −8.824084623150722613252793313114, −8.603099615747876676473975493991, −7.898330417445382248980147621505, −7.43230277029857419057003768353, −7.40855873304437277279336321533, −6.68778168822396917467510226930, −6.30803203395439759784562414877, −6.10657765522389319902276956235, −5.28621874690842429323146325444, −4.80757806477824896988595153503, −4.75719329024931057062271170034, −4.20375464771236087929042708642, −3.56734860828207284900353344800, −2.87443102788451217377524394499, −2.85085708624874405759539496436, −2.32741617231247832729762673611, −1.58311419672719928503560002456, −1.03848987213083492164566398181, −0.63204927302414236720041503084, 0.63204927302414236720041503084, 1.03848987213083492164566398181, 1.58311419672719928503560002456, 2.32741617231247832729762673611, 2.85085708624874405759539496436, 2.87443102788451217377524394499, 3.56734860828207284900353344800, 4.20375464771236087929042708642, 4.75719329024931057062271170034, 4.80757806477824896988595153503, 5.28621874690842429323146325444, 6.10657765522389319902276956235, 6.30803203395439759784562414877, 6.68778168822396917467510226930, 7.40855873304437277279336321533, 7.43230277029857419057003768353, 7.898330417445382248980147621505, 8.603099615747876676473975493991, 8.824084623150722613252793313114, 9.020415492340436091207147850099

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