Properties

Label 4-80e4-1.1-c1e2-0-0
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  − 8·17-s + 16·29-s − 8·37-s − 16·41-s − 8·49-s − 16·53-s + 12·61-s − 8·73-s − 9·81-s + 4·89-s + 8·97-s + 16·101-s − 20·109-s + 32·113-s + 2·121-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 + ⋯
L(s)  = 1  − 1.94·17-s + 2.97·29-s − 1.31·37-s − 2.49·41-s − 8/7·49-s − 2.19·53-s + 1.53·61-s − 0.936·73-s − 81-s + 0.423·89-s + 0.812·97-s + 1.59·101-s − 1.91·109-s + 3.01·113-s + 2/11·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7520697990\)
\(L(\frac12)\) \(\approx\) \(0.7520697990\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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

−8.365555142726693023650831452123, −7.938712579674749341027435627717, −7.42282764113718541422330001012, −7.04640742334732456526016431617, −6.71136399842423317635684107078, −6.56536775431522621880388433485, −6.14146914088126881397506072413, −5.91642868261680090152416654750, −5.05492272558939153819359159147, −4.90106940327372207356123000447, −4.78501527214180156637575818533, −4.35982032707162211369978842664, −3.61287328949256575702433709953, −3.56119392165859084201390689793, −2.86439867737768881148439064396, −2.65592909961908986369534758015, −1.99604304183357913283386804354, −1.66680322850065496209100505128, −1.08127268940309701371413169651, −0.22620076627417607289201011686, 0.22620076627417607289201011686, 1.08127268940309701371413169651, 1.66680322850065496209100505128, 1.99604304183357913283386804354, 2.65592909961908986369534758015, 2.86439867737768881148439064396, 3.56119392165859084201390689793, 3.61287328949256575702433709953, 4.35982032707162211369978842664, 4.78501527214180156637575818533, 4.90106940327372207356123000447, 5.05492272558939153819359159147, 5.91642868261680090152416654750, 6.14146914088126881397506072413, 6.56536775431522621880388433485, 6.71136399842423317635684107078, 7.04640742334732456526016431617, 7.42282764113718541422330001012, 7.938712579674749341027435627717, 8.365555142726693023650831452123

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