Properties

Label 4-768e2-1.1-c3e2-0-29
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $2053.31$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 27·9-s + 96·11-s + 108·17-s − 8·19-s − 238·25-s + 108·27-s + 576·33-s + 588·41-s + 376·43-s − 98·49-s + 648·51-s − 48·57-s + 504·59-s + 1.25e3·67-s + 2.01e3·73-s − 1.42e3·75-s + 405·81-s − 1.44e3·83-s + 2.96e3·89-s + 3.64e3·97-s + 2.59e3·99-s − 2.37e3·107-s − 780·113-s + 4.25e3·121-s + 3.52e3·123-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 2.63·11-s + 1.54·17-s − 0.0965·19-s − 1.90·25-s + 0.769·27-s + 3.03·33-s + 2.23·41-s + 1.33·43-s − 2/7·49-s + 1.77·51-s − 0.111·57-s + 1.11·59-s + 2.29·67-s + 3.22·73-s − 2.19·75-s + 5/9·81-s − 1.90·83-s + 3.53·89-s + 3.81·97-s + 2.63·99-s − 2.14·107-s − 0.649·113-s + 3.19·121-s + 2.58·123-s + 0.000698·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2053.31\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 589824,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.178406562\)
\(L(\frac12)\) \(\approx\) \(8.178406562\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
good5$C_2^2$ \( 1 + 238 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 48 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 2666 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 54 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 5666 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 22270 T^{2} + p^{6} T^{4} \)
31$C_2^2$ \( 1 + 56114 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 4726 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 294 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 188 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 48146 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 256946 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 252 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 445850 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 628 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 715774 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 1006 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 811150 T^{2} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 720 T + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 1482 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1822 T + p^{3} T^{2} )^{2} \)
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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

−9.864213414231707513735628758823, −9.585777157101643877839423888261, −9.232139199266545057078776861472, −9.127554318090184601681127317662, −8.402222838822267796650617855968, −7.993642063733579301703699665021, −7.65492739301406495932481225119, −7.27034818232677890748115266246, −6.57873635333717367301621435925, −6.34838347596536759967615958940, −5.79600529257246557483843267720, −5.27537782648544382172170985020, −4.41940794080689230665277687241, −4.03561332930643849414782087242, −3.55236082540374648393595097664, −3.46904425121685817270166890375, −2.26592253153613633575974834015, −2.11401839644373160514748643196, −1.05397915992800364683052286641, −0.916964568446941825513274142864, 0.916964568446941825513274142864, 1.05397915992800364683052286641, 2.11401839644373160514748643196, 2.26592253153613633575974834015, 3.46904425121685817270166890375, 3.55236082540374648393595097664, 4.03561332930643849414782087242, 4.41940794080689230665277687241, 5.27537782648544382172170985020, 5.79600529257246557483843267720, 6.34838347596536759967615958940, 6.57873635333717367301621435925, 7.27034818232677890748115266246, 7.65492739301406495932481225119, 7.993642063733579301703699665021, 8.402222838822267796650617855968, 9.127554318090184601681127317662, 9.232139199266545057078776861472, 9.585777157101643877839423888261, 9.864213414231707513735628758823

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