Properties

Label 4-80e4-1.1-c1e2-0-14
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  + 8·7-s + 9-s − 6·17-s + 8·23-s − 8·31-s + 10·41-s + 16·47-s + 34·49-s + 8·63-s + 16·71-s + 14·73-s − 8·79-s − 8·81-s + 2·89-s − 4·97-s + 16·103-s − 30·113-s − 48·119-s − 15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + ⋯
L(s)  = 1  + 3.02·7-s + 1/3·9-s − 1.45·17-s + 1.66·23-s − 1.43·31-s + 1.56·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s + 1.89·71-s + 1.63·73-s − 0.900·79-s − 8/9·81-s + 0.211·89-s − 0.406·97-s + 1.57·103-s − 2.82·113-s − 4.40·119-s − 1.36·121-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 − 0.485·153-s + 0.0798·157-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

\(L(1)\) \(\approx\) \(5.569555243\)
\(L(\frac12)\) \(\approx\) \(5.569555243\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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

−8.162475697531223146269450891657, −7.82850657812127283405008972941, −7.51912797924572778889452809468, −7.27777570293169737795473330128, −6.88697473407551430949172026276, −6.60519840329478162437117870180, −5.93677057846133888983778425086, −5.57910313558857586874057516929, −5.29422744765153280104290902466, −4.88309594770335573333794249793, −4.71870388229105096542154952376, −4.29431294743316650219918917896, −3.92680933686811581977002060459, −3.65316871584912584481991525708, −2.59315746224210677790079331192, −2.57435859360292555237855641968, −2.00486353130738188885043977501, −1.59662718783930337418300270791, −1.14698927854776593189027082004, −0.63739428570904934513409823509, 0.63739428570904934513409823509, 1.14698927854776593189027082004, 1.59662718783930337418300270791, 2.00486353130738188885043977501, 2.57435859360292555237855641968, 2.59315746224210677790079331192, 3.65316871584912584481991525708, 3.92680933686811581977002060459, 4.29431294743316650219918917896, 4.71870388229105096542154952376, 4.88309594770335573333794249793, 5.29422744765153280104290902466, 5.57910313558857586874057516929, 5.93677057846133888983778425086, 6.60519840329478162437117870180, 6.88697473407551430949172026276, 7.27777570293169737795473330128, 7.51912797924572778889452809468, 7.82850657812127283405008972941, 8.162475697531223146269450891657

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