Properties

Label 4-1134e2-1.1-c1e2-0-46
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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-s + 4·5-s + 7-s − 8-s + 4·10-s − 4·11-s − 3·13-s + 14-s − 16-s + 14·17-s + 4·19-s − 4·22-s − 23-s + 5·25-s − 3·26-s + 29-s + 9·31-s + 14·34-s + 4·35-s + 4·37-s + 4·38-s − 4·40-s + 6·41-s − 11·43-s − 46-s − 6·47-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.78·5-s + 0.377·7-s − 0.353·8-s + 1.26·10-s − 1.20·11-s − 0.832·13-s + 0.267·14-s − 1/4·16-s + 3.39·17-s + 0.917·19-s − 0.852·22-s − 0.208·23-s + 25-s − 0.588·26-s + 0.185·29-s + 1.61·31-s + 2.40·34-s + 0.676·35-s + 0.657·37-s + 0.648·38-s − 0.632·40-s + 0.937·41-s − 1.67·43-s − 0.147·46-s − 0.875·47-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1134} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.802720758\)
\(L(\frac12)\) \(\approx\) \(4.802720758\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−9.986610220033794347797949390537, −9.688055252582169242202201774361, −9.599529710541291309318814419239, −8.801838558211958905449470513163, −8.239519684841063398445548670316, −8.009912109215036566181079373788, −7.37820240309185472675763817588, −7.34541665730863228854204897368, −6.43872834135868752387953978128, −5.99156284188357886003041121367, −5.54863008499608325787317769441, −5.54277074959891702549547605585, −4.93274146410361059778964858225, −4.71658619666695023926200615945, −3.79669687165690145606757265958, −3.21918508835012727468601819835, −2.77749492408389771388729409347, −2.37360839683253855335232205117, −1.53237805723991942582043450396, −0.930254527108876132836615445688, 0.930254527108876132836615445688, 1.53237805723991942582043450396, 2.37360839683253855335232205117, 2.77749492408389771388729409347, 3.21918508835012727468601819835, 3.79669687165690145606757265958, 4.71658619666695023926200615945, 4.93274146410361059778964858225, 5.54277074959891702549547605585, 5.54863008499608325787317769441, 5.99156284188357886003041121367, 6.43872834135868752387953978128, 7.34541665730863228854204897368, 7.37820240309185472675763817588, 8.009912109215036566181079373788, 8.239519684841063398445548670316, 8.801838558211958905449470513163, 9.599529710541291309318814419239, 9.688055252582169242202201774361, 9.986610220033794347797949390537

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