Properties

Label 4-119e2-1.1-c0e2-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.1603281878\)
\(L(\frac12)\) \(\approx\) \(0.1603281878\)
\(L(1)\) 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

−14.04573251757204190644351240073, −13.65194338558675342126772884310, −12.68729914443010867183501235312, −12.04645147468382634720510609770, −11.84702716241586137927181515730, −11.44893583765498948948023432146, −10.99785240360884651423191253530, −10.44337523099519291188430246284, −9.878008452330803346119504481607, −9.197151922943741932582102142604, −8.439959401615032609354256443612, −8.252672509741494813962148682655, −7.59622314204297360302731343397, −7.30241031392590951611595560944, −6.12094459230689769919386566111, −5.48989736619258142965789236319, −4.97597257629778139191185706419, −4.26497223589162521783933658687, −3.27448017805867294444555410081, −1.52402899467920246594218058938, 1.52402899467920246594218058938, 3.27448017805867294444555410081, 4.26497223589162521783933658687, 4.97597257629778139191185706419, 5.48989736619258142965789236319, 6.12094459230689769919386566111, 7.30241031392590951611595560944, 7.59622314204297360302731343397, 8.252672509741494813962148682655, 8.439959401615032609354256443612, 9.197151922943741932582102142604, 9.878008452330803346119504481607, 10.44337523099519291188430246284, 10.99785240360884651423191253530, 11.44893583765498948948023432146, 11.84702716241586137927181515730, 12.04645147468382634720510609770, 12.68729914443010867183501235312, 13.65194338558675342126772884310, 14.04573251757204190644351240073

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