Properties

Label 4-9016e2-1.1-c1e2-0-4
Degree $4$
Conductor $81288256$
Sign $1$
Analytic cond. $5183.00$
Root an. cond. $8.48487$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·9-s − 8·11-s + 2·23-s − 2·25-s − 12·29-s + 4·37-s − 4·53-s + 8·67-s + 16·71-s − 8·79-s + 7·81-s + 32·99-s − 32·107-s + 20·109-s − 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + ⋯
L(s)  = 1  − 4/3·9-s − 2.41·11-s + 0.417·23-s − 2/5·25-s − 2.22·29-s + 0.657·37-s − 0.549·53-s + 0.977·67-s + 1.89·71-s − 0.900·79-s + 7/9·81-s + 3.21·99-s − 3.09·107-s + 1.91·109-s − 1.12·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81288256\)    =    \(2^{6} \cdot 7^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(5183.00\)
Root analytic conductor: \(8.48487\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81288256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
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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

−7.52366435033061348762004419079, −7.46017474460956941305015808769, −6.75921626579394268775695391022, −6.73541756931540007386576299256, −5.92293494997900362509171600771, −5.77468835733657028784887044682, −5.57776413903196387731037837172, −5.26626561456572696882026725712, −4.78074570109879279020150836950, −4.65929335200259489643422500671, −3.84640506870763569676600522155, −3.67994408394886647184093229120, −3.16097880805800157233664023576, −2.82280013416057160688853688980, −2.30484830344729372423548778208, −2.29689771563426668967331077590, −1.58520770994493866613355433740, −0.815005231937088310239810377387, 0, 0, 0.815005231937088310239810377387, 1.58520770994493866613355433740, 2.29689771563426668967331077590, 2.30484830344729372423548778208, 2.82280013416057160688853688980, 3.16097880805800157233664023576, 3.67994408394886647184093229120, 3.84640506870763569676600522155, 4.65929335200259489643422500671, 4.78074570109879279020150836950, 5.26626561456572696882026725712, 5.57776413903196387731037837172, 5.77468835733657028784887044682, 5.92293494997900362509171600771, 6.73541756931540007386576299256, 6.75921626579394268775695391022, 7.46017474460956941305015808769, 7.52366435033061348762004419079

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