Properties

Label 4-980e2-1.1-c1e2-0-21
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  + 3-s − 5-s + 3·9-s + 11-s + 10·13-s − 15-s + 17-s − 6·19-s + 4·23-s + 8·27-s + 6·29-s + 2·31-s + 33-s − 8·37-s + 10·39-s + 20·41-s − 4·43-s − 3·45-s − 7·47-s + 51-s + 2·53-s − 55-s − 6·57-s + 14·59-s − 8·61-s − 10·65-s − 14·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 9-s + 0.301·11-s + 2.77·13-s − 0.258·15-s + 0.242·17-s − 1.37·19-s + 0.834·23-s + 1.53·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s − 1.31·37-s + 1.60·39-s + 3.12·41-s − 0.609·43-s − 0.447·45-s − 1.02·47-s + 0.140·51-s + 0.274·53-s − 0.134·55-s − 0.794·57-s + 1.82·59-s − 1.02·61-s − 1.24·65-s − 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.353962477\)
\(L(\frac12)\) \(\approx\) \(3.353962477\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
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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

−10.40511142129460231541305498386, −9.789927330029288171510856477396, −9.202053634679022763824735940935, −8.789272231445184092629170714700, −8.590140956839011974167373014814, −8.345606578897386903466460853075, −7.72279916913196250510551119877, −7.37178789505501947467188505608, −6.61989014358559293073065467617, −6.55081251014903475283625746215, −6.07071845619069716537708830873, −5.55757060813207654062647818198, −4.57731960000913253770588061262, −4.55342840324492373718308685332, −3.84901127216902779642437630539, −3.53332478968547246071695883085, −2.98478980778539671612450640472, −2.24843793034550670062404991118, −1.35421397992367284149744390398, −0.985516337670595522033698410727, 0.985516337670595522033698410727, 1.35421397992367284149744390398, 2.24843793034550670062404991118, 2.98478980778539671612450640472, 3.53332478968547246071695883085, 3.84901127216902779642437630539, 4.55342840324492373718308685332, 4.57731960000913253770588061262, 5.55757060813207654062647818198, 6.07071845619069716537708830873, 6.55081251014903475283625746215, 6.61989014358559293073065467617, 7.37178789505501947467188505608, 7.72279916913196250510551119877, 8.345606578897386903466460853075, 8.590140956839011974167373014814, 8.789272231445184092629170714700, 9.202053634679022763824735940935, 9.789927330029288171510856477396, 10.40511142129460231541305498386

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