Properties

Label 4-320e2-1.1-c1e2-0-9
Degree $4$
Conductor $102400$
Sign $1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
Motivic weight $1$
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  − 2·5-s + 2·9-s + 4·13-s + 4·17-s + 3·25-s − 12·29-s + 20·37-s + 4·41-s − 4·45-s − 6·49-s − 12·53-s + 4·61-s − 8·65-s − 12·73-s − 5·81-s − 8·85-s + 20·89-s + 4·97-s + 4·101-s + 36·109-s + 4·113-s + 8·117-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s + 2/3·9-s + 1.10·13-s + 0.970·17-s + 3/5·25-s − 2.22·29-s + 3.28·37-s + 0.624·41-s − 0.596·45-s − 6/7·49-s − 1.64·53-s + 0.512·61-s − 0.992·65-s − 1.40·73-s − 5/9·81-s − 0.867·85-s + 2.11·89-s + 0.406·97-s + 0.398·101-s + 3.44·109-s + 0.376·113-s + 0.739·117-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(102400\)    =    \(2^{12} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(6.52911\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437687236\)
\(L(\frac12)\) \(\approx\) \(1.437687236\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−11.54096329063596514712291905022, −11.53817667191451373632568628568, −11.03111970956455570875612605421, −10.63652798535804361642446776952, −9.862515474013922215550479093911, −9.626869907218388433642260600245, −9.096978939488460814703141204352, −8.519721980105615848195832680186, −7.943670102081038157537312274234, −7.48765597332836045376350376470, −7.42867144191638870823373674443, −6.38912401683433574009480666981, −6.06943653388910276154682339658, −5.51245569181223824742894392200, −4.59789173031085180214409789496, −4.27226961306955603738878899232, −3.54943286150140438787827956806, −3.12925247737475439352160521810, −1.95006760437674357452119030431, −0.942019062778363645446364015230, 0.942019062778363645446364015230, 1.95006760437674357452119030431, 3.12925247737475439352160521810, 3.54943286150140438787827956806, 4.27226961306955603738878899232, 4.59789173031085180214409789496, 5.51245569181223824742894392200, 6.06943653388910276154682339658, 6.38912401683433574009480666981, 7.42867144191638870823373674443, 7.48765597332836045376350376470, 7.943670102081038157537312274234, 8.519721980105615848195832680186, 9.096978939488460814703141204352, 9.626869907218388433642260600245, 9.862515474013922215550479093911, 10.63652798535804361642446776952, 11.03111970956455570875612605421, 11.53817667191451373632568628568, 11.54096329063596514712291905022

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