Properties

Degree $4$
Conductor $88510464$
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 − 8·15-s + 4·17-s − 8·19-s − 4·23-s + 4·25-s − 4·27-s + 16·31-s − 8·33-s + 8·37-s + 4·41-s − 16·43-s + 12·45-s + 8·47-s − 8·51-s + 4·53-s + 16·55-s + 16·57-s − 16·59-s + 8·61-s − 16·67-s + 8·69-s − 4·71-s + 8·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s − 2.06·15-s + 0.970·17-s − 1.83·19-s − 0.834·23-s + 4/5·25-s − 0.769·27-s + 2.87·31-s − 1.39·33-s + 1.31·37-s + 0.624·41-s − 2.43·43-s + 1.78·45-s + 1.16·47-s − 1.12·51-s + 0.549·53-s + 2.15·55-s + 2.11·57-s − 2.08·59-s + 1.02·61-s − 1.95·67-s + 0.963·69-s − 0.474·71-s + 0.936·73-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

Degree: \(4\)
Conductor: \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.047588157\)
\(L(\frac12)\) \(\approx\) \(3.047588157\)
\(L(\frac{3}{2})\) 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)$
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$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 16 T + 118 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.68742066106454099519409554696, −7.59754952493918638532494705160, −6.98749367846939955155565705517, −6.62742805603042708442395695884, −6.20615407374344862714351527046, −6.19337992088851168134332544181, −5.98701100628049734013957454183, −5.79716131908504674783806117743, −4.96802595292866187351412874698, −4.93439455815952697113790098947, −4.45433284318597008617127379976, −4.24688682768405466642746601112, −3.63303687257312336016518181841, −3.35061619128723922666464315328, −2.58501180612700446764246689877, −2.30101040335878366305906454952, −1.92645126436223169599472639519, −1.37358040504496856404541466744, −1.10102610468143054209071671379, −0.45180679034758887988236174591, 0.45180679034758887988236174591, 1.10102610468143054209071671379, 1.37358040504496856404541466744, 1.92645126436223169599472639519, 2.30101040335878366305906454952, 2.58501180612700446764246689877, 3.35061619128723922666464315328, 3.63303687257312336016518181841, 4.24688682768405466642746601112, 4.45433284318597008617127379976, 4.93439455815952697113790098947, 4.96802595292866187351412874698, 5.79716131908504674783806117743, 5.98701100628049734013957454183, 6.19337992088851168134332544181, 6.20615407374344862714351527046, 6.62742805603042708442395695884, 6.98749367846939955155565705517, 7.59754952493918638532494705160, 7.68742066106454099519409554696

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