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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 2·7-s − 4·13-s + 12·16-s − 12·19-s + 3·25-s − 8·28-s − 4·31-s − 4·37-s + 10·43-s + 3·49-s + 16·52-s + 28·61-s − 32·64-s − 30·67-s − 8·73-s + 48·76-s − 8·91-s − 16·97-s − 12·100-s + 12·103-s − 14·109-s + 24·112-s + 16·124-s + 127-s + 131-s − 24·133-s + ⋯
L(s)  = 1  − 2·4-s + 0.755·7-s − 1.10·13-s + 3·16-s − 2.75·19-s + 3/5·25-s − 1.51·28-s − 0.718·31-s − 0.657·37-s + 1.52·43-s + 3/7·49-s + 2.21·52-s + 3.58·61-s − 4·64-s − 3.66·67-s − 0.936·73-s + 5.50·76-s − 0.838·91-s − 1.62·97-s − 6/5·100-s + 1.18·103-s − 1.34·109-s + 2.26·112-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-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$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−7.74548315788270244442390894623, −7.46642602553897603878874928180, −7.04983755009916007187010712185, −6.79434561963590601420994594175, −6.07649167917037921142061042238, −5.95658369521250100485905017944, −5.42321335571078752195943784833, −5.24036065205136004085315406640, −4.77037693505450367401066851904, −4.45131399783077334962623494481, −4.15983607887869850194494891195, −4.09141912673313470117227320076, −3.42165639423045718245830645924, −3.00635846342101203894864249143, −2.32164320616891640454317839390, −2.13003057079758333884469944987, −1.40552508185698653527656973107, −0.904918320736879303599730197489, 0, 0, 0.904918320736879303599730197489, 1.40552508185698653527656973107, 2.13003057079758333884469944987, 2.32164320616891640454317839390, 3.00635846342101203894864249143, 3.42165639423045718245830645924, 4.09141912673313470117227320076, 4.15983607887869850194494891195, 4.45131399783077334962623494481, 4.77037693505450367401066851904, 5.24036065205136004085315406640, 5.42321335571078752195943784833, 5.95658369521250100485905017944, 6.07649167917037921142061042238, 6.79434561963590601420994594175, 7.04983755009916007187010712185, 7.46642602553897603878874928180, 7.74548315788270244442390894623

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