Properties

Degree 4
Conductor $ 3^{4} \cdot 7^{2} \cdot 11^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4·4-s − 2·5-s − 2·7-s − 3·8-s + 6·10-s − 2·13-s + 6·14-s + 3·16-s − 5·17-s + 8·19-s − 8·20-s + 23-s − 7·25-s + 6·26-s − 8·28-s − 7·29-s − 4·31-s − 6·32-s + 15·34-s + 4·35-s + 4·37-s − 24·38-s + 6·40-s + 5·43-s − 3·46-s + 11·47-s + ⋯
L(s)  = 1  − 2.12·2-s + 2·4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s + 1.89·10-s − 0.554·13-s + 1.60·14-s + 3/4·16-s − 1.21·17-s + 1.83·19-s − 1.78·20-s + 0.208·23-s − 7/5·25-s + 1.17·26-s − 1.51·28-s − 1.29·29-s − 0.718·31-s − 1.06·32-s + 2.57·34-s + 0.676·35-s + 0.657·37-s − 3.89·38-s + 0.948·40-s + 0.762·43-s − 0.442·46-s + 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(58110129\)    =    \(3^{4} \cdot 7^{2} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 58110129,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 49 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 11 T + 123 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 11 T + 117 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 13 T + 163 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 21 T + 221 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 24 T + 285 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 15 T + 213 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 179 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.76154565758793658606021479162, −7.48085905236345556870915698685, −7.17704354450408090541549292219, −7.03150306474627187014297388811, −6.60010221302847682970177704701, −5.90075080593352903463373853332, −5.68268472682474777385594792950, −5.57970956095861238270863247684, −4.75039439926802169622314186159, −4.57890786804700912309586076732, −3.82840143272932226708281518862, −3.66468988033671951088633000857, −3.44217108486327928255452659176, −2.70026555537369215000132438275, −2.15715976539399520472391528167, −2.10319146656435429834705392838, −1.02296854151087814089041589982, −0.911068768582631026107892254663, 0, 0, 0.911068768582631026107892254663, 1.02296854151087814089041589982, 2.10319146656435429834705392838, 2.15715976539399520472391528167, 2.70026555537369215000132438275, 3.44217108486327928255452659176, 3.66468988033671951088633000857, 3.82840143272932226708281518862, 4.57890786804700912309586076732, 4.75039439926802169622314186159, 5.57970956095861238270863247684, 5.68268472682474777385594792950, 5.90075080593352903463373853332, 6.60010221302847682970177704701, 7.03150306474627187014297388811, 7.17704354450408090541549292219, 7.48085905236345556870915698685, 7.76154565758793658606021479162

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