Properties

Label 4-159e2-1.1-c0e2-0-1
Degree $4$
Conductor $25281$
Sign $1$
Analytic cond. $0.00629663$
Root an. cond. $0.281693$
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 + 2·3-s − 5-s − 2·6-s − 7-s + 3·9-s + 10-s − 13-s + 14-s − 2·15-s − 3·18-s − 2·21-s − 23-s + 26-s + 4·27-s + 2·30-s + 32-s + 35-s − 37-s − 2·39-s − 41-s + 2·42-s − 43-s − 3·45-s + 46-s + 2·53-s − 4·54-s + ⋯
L(s)  = 1  − 2-s + 2·3-s − 5-s − 2·6-s − 7-s + 3·9-s + 10-s − 13-s + 14-s − 2·15-s − 3·18-s − 2·21-s − 23-s + 26-s + 4·27-s + 2·30-s + 32-s + 35-s − 37-s − 2·39-s − 41-s + 2·42-s − 43-s − 3·45-s + 46-s + 2·53-s − 4·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25281\)    =    \(3^{2} \cdot 53^{2}\)
Sign: $1$
Analytic conductor: \(0.00629663\)
Root analytic conductor: \(0.281693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{159} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25281,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3491217167\)
\(L(\frac12)\) \(\approx\) \(0.3491217167\)
\(L(1)\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
53$C_1$ \( ( 1 - T )^{2} \)
good2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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

−13.29794649973420546914448632559, −13.12971081393997962784196767074, −12.31912059848638155757535555552, −12.08075452607637885321168741455, −11.50962232126074463883831350213, −10.37181457340143541192994701428, −9.976018845523336640746641565715, −9.929638345352002043042230505657, −9.140744998868018239527572375951, −8.827944736570786066227315986209, −8.319312957477228354448347977821, −7.985342886408146721932302617208, −7.16533614001499362983021333869, −7.12455000812335815902247706939, −6.18693376072904024118256695995, −4.92042080695935339238281285711, −4.19516455922442619181815882044, −3.56732910811294467104717462395, −2.99525620404322713807727056053, −2.06209860374741796339889994643, 2.06209860374741796339889994643, 2.99525620404322713807727056053, 3.56732910811294467104717462395, 4.19516455922442619181815882044, 4.92042080695935339238281285711, 6.18693376072904024118256695995, 7.12455000812335815902247706939, 7.16533614001499362983021333869, 7.985342886408146721932302617208, 8.319312957477228354448347977821, 8.827944736570786066227315986209, 9.140744998868018239527572375951, 9.929638345352002043042230505657, 9.976018845523336640746641565715, 10.37181457340143541192994701428, 11.50962232126074463883831350213, 12.08075452607637885321168741455, 12.31912059848638155757535555552, 13.12971081393997962784196767074, 13.29794649973420546914448632559

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