Properties

Label 4-131e2-1.1-c0e2-0-0
Degree $4$
Conductor $17161$
Sign $1$
Analytic cond. $0.00427421$
Root an. cond. $0.255690$
Motivic weight $0$
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  − 3-s + 2·4-s − 5-s − 7-s − 11-s − 2·12-s − 13-s + 15-s + 3·16-s − 2·20-s + 21-s − 2·28-s + 33-s + 35-s + 39-s − 41-s − 43-s − 2·44-s − 3·48-s − 2·52-s + 4·53-s + 55-s − 59-s + 2·60-s − 61-s + 4·64-s + 65-s + ⋯
L(s)  = 1  − 3-s + 2·4-s − 5-s − 7-s − 11-s − 2·12-s − 13-s + 15-s + 3·16-s − 2·20-s + 21-s − 2·28-s + 33-s + 35-s + 39-s − 41-s − 43-s − 2·44-s − 3·48-s − 2·52-s + 4·53-s + 55-s − 59-s + 2·60-s − 61-s + 4·64-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17161\)    =    \(131^{2}\)
Sign: $1$
Analytic conductor: \(0.00427421\)
Root analytic conductor: \(0.255690\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{131} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17161,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3263700811\)
\(L(\frac12)\) \(\approx\) \(0.3263700811\)
\(L(1)\) 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)$
bad131$C_1$ \( ( 1 - T )^{2} \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
3$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
13$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
43$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$ \( ( 1 - T )^{4} \)
59$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
61$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−13.49115710230681409798432755353, −13.22375212491396362504805975671, −12.29259752204418281575577204325, −12.21274170830972099991167298192, −11.66281031150107468242863614059, −11.55149329870096178406183095752, −10.63275133986296812825530566576, −10.50955681500881916104758058698, −10.04012334184021374516029299482, −9.218449254119487021935505079960, −8.157385197801971307650196111733, −7.81691846039840170699782085388, −7.11096588269597252147591005558, −6.88405597292774965520044607618, −6.14622486610113318867757722039, −5.62217519851476845691626484153, −5.02577259089436039660102904084, −3.75376533239769561770804369848, −3.03544317976724495517564615348, −2.26288523295514045403412685904, 2.26288523295514045403412685904, 3.03544317976724495517564615348, 3.75376533239769561770804369848, 5.02577259089436039660102904084, 5.62217519851476845691626484153, 6.14622486610113318867757722039, 6.88405597292774965520044607618, 7.11096588269597252147591005558, 7.81691846039840170699782085388, 8.157385197801971307650196111733, 9.218449254119487021935505079960, 10.04012334184021374516029299482, 10.50955681500881916104758058698, 10.63275133986296812825530566576, 11.55149329870096178406183095752, 11.66281031150107468242863614059, 12.21274170830972099991167298192, 12.29259752204418281575577204325, 13.22375212491396362504805975671, 13.49115710230681409798432755353

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