Properties

Degree 4
Conductor $ 3^{4} \cdot 7^{2} \cdot 11^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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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  =  0
Selberg data  =  $(4,\ 58110129,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $7.697735644$
$L(\frac12)$  $\approx$  $7.697735644$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.908194373865417330099139044418, −7.68934159419656972091396200603, −7.37166020921835813166324596170, −6.92586723228949481823625754002, −6.29298224933793876586478225372, −6.27358903097402195513797088045, −5.80759678806505477048017754202, −5.58997789516078427970213017297, −5.07185891220514493673195283124, −4.82053314484135670136854997052, −4.37011139733133945289323511327, −4.24324501819282769733137107586, −3.77890911142711256929024663932, −3.66572637147918686773064194373, −3.08612920668107643081192969018, −2.79884274951301107545149716280, −2.09908141979025997299888366865, −1.80386955893955077603660031467, −1.13051217297041408005331222061, −0.47476446156402767335441390725, 0.47476446156402767335441390725, 1.13051217297041408005331222061, 1.80386955893955077603660031467, 2.09908141979025997299888366865, 2.79884274951301107545149716280, 3.08612920668107643081192969018, 3.66572637147918686773064194373, 3.77890911142711256929024663932, 4.24324501819282769733137107586, 4.37011139733133945289323511327, 4.82053314484135670136854997052, 5.07185891220514493673195283124, 5.58997789516078427970213017297, 5.80759678806505477048017754202, 6.27358903097402195513797088045, 6.29298224933793876586478225372, 6.92586723228949481823625754002, 7.37166020921835813166324596170, 7.68934159419656972091396200603, 7.908194373865417330099139044418

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