Properties

Label 4-1815e2-1.1-c1e2-0-38
Degree $4$
Conductor $3294225$
Sign $-1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·4-s + 5-s + 9-s + 4·12-s + 2·15-s + 2·20-s − 5·23-s − 4·25-s − 4·27-s + 10·31-s + 2·36-s − 14·37-s + 45-s − 7·47-s + 2·49-s + 2·53-s + 9·59-s + 4·60-s − 8·64-s − 18·67-s − 10·69-s − 3·71-s − 8·75-s − 11·81-s + 15·89-s − 10·92-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s + 0.447·5-s + 1/3·9-s + 1.15·12-s + 0.516·15-s + 0.447·20-s − 1.04·23-s − 4/5·25-s − 0.769·27-s + 1.79·31-s + 1/3·36-s − 2.30·37-s + 0.149·45-s − 1.02·47-s + 2/7·49-s + 0.274·53-s + 1.17·59-s + 0.516·60-s − 64-s − 2.19·67-s − 1.20·69-s − 0.356·71-s − 0.923·75-s − 1.22·81-s + 1.58·89-s − 1.04·92-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.32174396110033485275252370987, −6.91874715076244543084735179757, −6.57196878115659805152469188001, −6.06237513520528007700414794351, −5.81818993572122600709980014977, −5.25604663573377280035369940112, −4.69151093880696795566685281695, −4.21177433863618048450109418602, −3.63266113428945224882450786268, −3.26631167624545286828943213403, −2.70395128959344637055620916381, −2.30030715119460572622255909463, −1.87410963207314380950584746636, −1.36352988561360373004612983076, 0, 1.36352988561360373004612983076, 1.87410963207314380950584746636, 2.30030715119460572622255909463, 2.70395128959344637055620916381, 3.26631167624545286828943213403, 3.63266113428945224882450786268, 4.21177433863618048450109418602, 4.69151093880696795566685281695, 5.25604663573377280035369940112, 5.81818993572122600709980014977, 6.06237513520528007700414794351, 6.57196878115659805152469188001, 6.91874715076244543084735179757, 7.32174396110033485275252370987

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