Properties

Label 4-1148e2-1.1-c0e2-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4878348991\)
\(L(\frac12)\) \(\approx\) \(0.4878348991\)
\(L(1)\) 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

−10.37321964676910319257055147918, −9.746447742330723522468512071800, −9.538139982747020315127533888076, −8.969755346835513017661856541051, −8.662417026945543986841090108095, −8.160775831827835085407142989910, −7.85290978032906959231405822599, −7.29045281181377241814306036779, −7.03541724368761814260918152495, −6.54556642574320883872358064586, −5.75550773140910433551476629299, −5.70085743290770348017164171451, −5.06437340421326642834258678823, −4.89034434730231048082670550569, −4.11489247448533858616863579555, −3.91173057400785838861138862363, −2.69498805889845761080913673024, −2.07001008651529878065695396188, −1.73217175561221329925690682096, −0.827225766858395063243323841899, 0.827225766858395063243323841899, 1.73217175561221329925690682096, 2.07001008651529878065695396188, 2.69498805889845761080913673024, 3.91173057400785838861138862363, 4.11489247448533858616863579555, 4.89034434730231048082670550569, 5.06437340421326642834258678823, 5.70085743290770348017164171451, 5.75550773140910433551476629299, 6.54556642574320883872358064586, 7.03541724368761814260918152495, 7.29045281181377241814306036779, 7.85290978032906959231405822599, 8.160775831827835085407142989910, 8.662417026945543986841090108095, 8.969755346835513017661856541051, 9.538139982747020315127533888076, 9.746447742330723522468512071800, 10.37321964676910319257055147918

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