Properties

Label 4-119e2-1.1-c0e2-0-1
Degree $4$
Conductor $14161$
Sign $1$
Analytic cond. $0.00352702$
Root an. cond. $0.243698$
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  − 2-s + 3-s + 5-s − 6-s − 2·7-s − 10-s + 2·14-s + 15-s − 2·17-s − 2·21-s − 30-s + 31-s + 32-s + 2·34-s − 2·35-s + 41-s + 2·42-s − 43-s + 3·49-s − 2·51-s − 53-s + 61-s − 62-s − 64-s − 67-s + 2·70-s + 73-s + ⋯
L(s)  = 1  − 2-s + 3-s + 5-s − 6-s − 2·7-s − 10-s + 2·14-s + 15-s − 2·17-s − 2·21-s − 30-s + 31-s + 32-s + 2·34-s − 2·35-s + 41-s + 2·42-s − 43-s + 3·49-s − 2·51-s − 53-s + 61-s − 62-s − 64-s − 67-s + 2·70-s + 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14161\)    =    \(7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.00352702\)
Root analytic conductor: \(0.243698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{119} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14161,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2583543231\)
\(L(\frac12)\) \(\approx\) \(0.2583543231\)
\(L(1)\) 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)$
bad7$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$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$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
67$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
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$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
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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

−13.78274557559068410539747469819, −13.60067471336295545512712433556, −13.00791308066877707094697030464, −12.81832341078413986102119274275, −11.97448861661235573494853187038, −11.32903053497152410652726386718, −10.52527424062377841113145891281, −10.06890896841210859876690174106, −9.578200214180807123496499712112, −9.307261695204681645728448675397, −8.770078937487255232414253344988, −8.562806141941940391751284480268, −7.68592225538036914165305710456, −6.75426791784900440825992303891, −6.42543460903241917673966198786, −5.93263218695135416564774589031, −4.77696768335113156866800341807, −3.81385230800766422131085280182, −2.86242758482639499100631131705, −2.36971271317674985350180148952, 2.36971271317674985350180148952, 2.86242758482639499100631131705, 3.81385230800766422131085280182, 4.77696768335113156866800341807, 5.93263218695135416564774589031, 6.42543460903241917673966198786, 6.75426791784900440825992303891, 7.68592225538036914165305710456, 8.562806141941940391751284480268, 8.770078937487255232414253344988, 9.307261695204681645728448675397, 9.578200214180807123496499712112, 10.06890896841210859876690174106, 10.52527424062377841113145891281, 11.32903053497152410652726386718, 11.97448861661235573494853187038, 12.81832341078413986102119274275, 13.00791308066877707094697030464, 13.60067471336295545512712433556, 13.78274557559068410539747469819

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