Properties

Label 4-801e2-1.1-c1e2-0-0
Degree $4$
Conductor $641601$
Sign $1$
Analytic cond. $40.9090$
Root an. cond. $2.52903$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 12·16-s − 10·25-s + 14·49-s − 32·64-s − 2·67-s + 10·73-s − 14·79-s + 22·97-s + 40·100-s − 26·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2·25-s + 2·49-s − 4·64-s − 0.244·67-s + 1.17·73-s − 1.57·79-s + 2.23·97-s + 4·100-s − 2.49·109-s − 2·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 641601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 641601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(641601\)    =    \(3^{4} \cdot 89^{2}\)
Sign: $1$
Analytic conductor: \(40.9090\)
Root analytic conductor: \(2.52903\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 641601,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7310814825\)
\(L(\frac12)\) \(\approx\) \(0.7310814825\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
89$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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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

−10.29782382212089117875433344330, −10.00892250392027775872273357288, −9.550771790573832378443027155069, −9.191423182313716890548511217156, −8.902200195777242438937755909023, −8.407279713197925052163265155535, −7.974908688946724448869699809349, −7.69045710638473315079962729984, −7.18561155073859624406852722121, −6.45827437551555765686664075317, −5.86917644712850163062864649701, −5.56199420487264253592771873444, −5.15783074656497583469979570467, −4.52330119533499255224262395432, −4.09806919249719650622794122193, −3.79261438973760602015649745895, −3.18616646491018548666489390246, −2.33055703087610667011952854258, −1.41052068632978185547053056838, −0.47170587173855267234238980133, 0.47170587173855267234238980133, 1.41052068632978185547053056838, 2.33055703087610667011952854258, 3.18616646491018548666489390246, 3.79261438973760602015649745895, 4.09806919249719650622794122193, 4.52330119533499255224262395432, 5.15783074656497583469979570467, 5.56199420487264253592771873444, 5.86917644712850163062864649701, 6.45827437551555765686664075317, 7.18561155073859624406852722121, 7.69045710638473315079962729984, 7.974908688946724448869699809349, 8.407279713197925052163265155535, 8.902200195777242438937755909023, 9.191423182313716890548511217156, 9.550771790573832378443027155069, 10.00892250392027775872273357288, 10.29782382212089117875433344330

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