Properties

Label 4-56e4-1.1-c1e2-0-20
Degree $4$
Conductor $9834496$
Sign $1$
Analytic cond. $627.055$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 8·11-s − 8·23-s − 8·25-s − 16·29-s + 16·37-s + 8·43-s − 20·53-s + 16·79-s − 5·81-s − 16·99-s − 16·107-s + 16·109-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 − 8·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.41·11-s − 1.66·23-s − 8/5·25-s − 2.97·29-s + 2.63·37-s + 1.21·43-s − 2.74·53-s + 1.80·79-s − 5/9·81-s − 1.60·99-s − 1.54·107-s + 1.53·109-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 − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9834496\)    =    \(2^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(627.055\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9834496,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 192 T^{2} + p^{2} T^{4} \)
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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

−8.124939593837381014927572600669, −8.036695708653894452443146779578, −7.79378619306368188405469241024, −7.50207714511441993113073894176, −7.32451516602910773036317892267, −6.50375889240135103698234272055, −6.08849465501517754617757115250, −5.82824166734359957270460478545, −5.57843676375437461370432574756, −5.01805210124390808189485564284, −4.65272111652703885058738402262, −4.18310334230384274108295725979, −3.74163182406624173881593430028, −3.44198429164211963672830386157, −2.57285819352033877359464065658, −2.37242599754759520820039395423, −1.95016533601375678919844923526, −1.25766049639397829647372456169, 0, 0, 1.25766049639397829647372456169, 1.95016533601375678919844923526, 2.37242599754759520820039395423, 2.57285819352033877359464065658, 3.44198429164211963672830386157, 3.74163182406624173881593430028, 4.18310334230384274108295725979, 4.65272111652703885058738402262, 5.01805210124390808189485564284, 5.57843676375437461370432574756, 5.82824166734359957270460478545, 6.08849465501517754617757115250, 6.50375889240135103698234272055, 7.32451516602910773036317892267, 7.50207714511441993113073894176, 7.79378619306368188405469241024, 8.036695708653894452443146779578, 8.124939593837381014927572600669

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