Properties

Label 4-80e4-1.1-c1e2-0-12
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·9-s + 6·11-s + 6·17-s − 2·19-s − 14·27-s + 12·33-s + 18·41-s + 8·43-s − 2·49-s + 12·51-s − 4·57-s − 24·59-s + 22·67-s − 14·73-s − 4·81-s + 30·83-s + 6·89-s + 28·97-s − 18·99-s + 18·107-s − 30·113-s + 5·121-s + 36·123-s + 127-s + 16·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s − 9-s + 1.80·11-s + 1.45·17-s − 0.458·19-s − 2.69·27-s + 2.08·33-s + 2.81·41-s + 1.21·43-s − 2/7·49-s + 1.68·51-s − 0.529·57-s − 3.12·59-s + 2.68·67-s − 1.63·73-s − 4/9·81-s + 3.29·83-s + 0.635·89-s + 2.84·97-s − 1.80·99-s + 1.74·107-s − 2.82·113-s + 5/11·121-s + 3.24·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(5.201384900\)
\(L(\frac12)\) \(\approx\) \(5.201384900\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−8.092215483860948624802294134786, −7.77103803940088625522562349480, −7.60334690251796714033122268217, −7.55075724730991410678530337332, −6.58313598318234900386278080485, −6.48762318905448830540571611019, −6.03131544470245267037028427810, −5.93646480763440410149302484187, −5.35539635736104099358611396187, −5.03995983879745872141396852651, −4.42917273509330128606505983426, −4.07577574982298080321030268756, −3.61789437003575579512958960590, −3.53982147367611561777893790859, −2.82141262863596428801118183360, −2.78210057972701347165061188507, −2.08181765179382447060208515786, −1.75753263798141382406855779200, −1.01051541151638899632887973399, −0.59761585589520308526573817323, 0.59761585589520308526573817323, 1.01051541151638899632887973399, 1.75753263798141382406855779200, 2.08181765179382447060208515786, 2.78210057972701347165061188507, 2.82141262863596428801118183360, 3.53982147367611561777893790859, 3.61789437003575579512958960590, 4.07577574982298080321030268756, 4.42917273509330128606505983426, 5.03995983879745872141396852651, 5.35539635736104099358611396187, 5.93646480763440410149302484187, 6.03131544470245267037028427810, 6.48762318905448830540571611019, 6.58313598318234900386278080485, 7.55075724730991410678530337332, 7.60334690251796714033122268217, 7.77103803940088625522562349480, 8.092215483860948624802294134786

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