Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 7^{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  + 2·3-s + 4·5-s + 3·9-s + 4·11-s + 8·13-s + 8·15-s − 4·17-s − 4·23-s + 4·25-s + 4·27-s + 8·29-s + 8·31-s + 8·33-s + 8·37-s + 16·39-s − 4·41-s + 12·45-s − 8·51-s + 4·53-s + 16·55-s − 8·59-s + 16·61-s + 32·65-s − 8·69-s − 4·71-s − 8·73-s + 8·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s + 2.21·13-s + 2.06·15-s − 0.970·17-s − 0.834·23-s + 4/5·25-s + 0.769·27-s + 1.48·29-s + 1.43·31-s + 1.39·33-s + 1.31·37-s + 2.56·39-s − 0.624·41-s + 1.78·45-s − 1.12·51-s + 0.549·53-s + 2.15·55-s − 1.04·59-s + 2.04·61-s + 3.96·65-s − 0.963·69-s − 0.474·71-s − 0.936·73-s + 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(13.61578216\)
\(L(\frac12)\)  \(\approx\)  \(13.61578216\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 168 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 64 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.927424926919604154160952321092, −7.72585036886932323220074391190, −7.03299938786236444141022589872, −6.72203030971948567910003152864, −6.42369736977622451905066485189, −6.25217416863931145683493452018, −5.92345305974184025535507635764, −5.77118004697093099328645255204, −5.01363930293116518144775743042, −4.72665967251638688545818683860, −4.19678897830106618602179062157, −4.10859726306704098474389389594, −3.41806302557826839464637869886, −3.41317056138765672993485839396, −2.64645742354211029806225230049, −2.42107459796500435031814111383, −1.86501110355699153164315223581, −1.73279940446944396283776078779, −1.00525718599102715798861008895, −0.889527180161340789107451627610, 0.889527180161340789107451627610, 1.00525718599102715798861008895, 1.73279940446944396283776078779, 1.86501110355699153164315223581, 2.42107459796500435031814111383, 2.64645742354211029806225230049, 3.41317056138765672993485839396, 3.41806302557826839464637869886, 4.10859726306704098474389389594, 4.19678897830106618602179062157, 4.72665967251638688545818683860, 5.01363930293116518144775743042, 5.77118004697093099328645255204, 5.92345305974184025535507635764, 6.25217416863931145683493452018, 6.42369736977622451905066485189, 6.72203030971948567910003152864, 7.03299938786236444141022589872, 7.72585036886932323220074391190, 7.927424926919604154160952321092

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