Properties

Label 4-1148e2-1.1-c0e2-0-1
Degree $4$
Conductor $1317904$
Sign $1$
Analytic cond. $0.328244$
Root an. cond. $0.756919$
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 − 7-s + 8-s + 9-s − 10-s + 11-s + 14-s + 15-s − 16-s − 18-s + 19-s − 21-s − 22-s + 24-s + 25-s + 2·27-s − 30-s + 33-s − 35-s − 2·37-s − 38-s + 40-s + 2·41-s + 42-s + ⋯
L(s)  = 1  − 2-s + 3-s + 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 14-s + 15-s − 16-s − 18-s + 19-s − 21-s − 22-s + 24-s + 25-s + 2·27-s − 30-s + 33-s − 35-s − 2·37-s − 38-s + 40-s + 2·41-s + 42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.328244\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1317904,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003880167\)
\(L(\frac12)\) \(\approx\) \(1.003880167\)
\(L(1)\) 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)$
bad2$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 + T + T^{2} )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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

−10.03449868006302145032090143066, −9.700112880000607070960059493996, −9.246254029239740980119232918786, −8.986290818922945568309886387979, −8.961563577808475987237338630687, −8.223806072998561826942099750806, −7.955594975165847919820021026106, −7.29191951928437660531015918005, −6.87715081433727202897705999294, −6.82588037278540464491630510377, −6.08151520254665653405581156349, −5.71640639534600054393975751587, −4.97588879065555881578862278760, −4.56925172812211455262890776688, −4.06596236240117556461458709393, −3.26204845692091433435079383733, −3.13573954958441692168331243465, −2.35238718539025944563366193926, −1.56960827996413838761891495272, −1.21075637368525368329866818591, 1.21075637368525368329866818591, 1.56960827996413838761891495272, 2.35238718539025944563366193926, 3.13573954958441692168331243465, 3.26204845692091433435079383733, 4.06596236240117556461458709393, 4.56925172812211455262890776688, 4.97588879065555881578862278760, 5.71640639534600054393975751587, 6.08151520254665653405581156349, 6.82588037278540464491630510377, 6.87715081433727202897705999294, 7.29191951928437660531015918005, 7.955594975165847919820021026106, 8.223806072998561826942099750806, 8.961563577808475987237338630687, 8.986290818922945568309886387979, 9.246254029239740980119232918786, 9.700112880000607070960059493996, 10.03449868006302145032090143066

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