Properties

Label 4-1776e2-1.1-c0e2-0-2
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $0.785597$
Root an. cond. $0.941456$
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  + 3-s + 7-s − 3·13-s + 21-s + 25-s − 27-s + 2·37-s − 3·39-s + 49-s + 67-s + 2·73-s + 75-s + 3·79-s − 81-s − 3·91-s − 3·109-s + 2·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 3-s + 7-s − 3·13-s + 21-s + 25-s − 27-s + 2·37-s − 3·39-s + 49-s + 67-s + 2·73-s + 75-s + 3·79-s − 81-s − 3·91-s − 3·109-s + 2·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + 163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.785597\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1776} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3154176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.563164211\)
\(L(\frac12)\) \(\approx\) \(1.563164211\)
\(L(1)\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−9.613009786047755710339410458238, −9.361676175737549223754440137754, −8.933022611431709029939379201112, −8.381170347648151226574497511031, −8.084263798150192620438116216965, −7.75157693361575687240855413046, −7.46381363676700004773407029293, −7.12753286825809124388889063578, −6.55378443777929638949203500807, −6.19271244943421759707244801744, −5.29213107875301605217683294288, −5.18283535932327000913724474644, −4.89250490777436438282645973149, −4.24042279354031696152018996599, −3.95239635753751879776209999070, −3.10741552901677517301310799421, −2.73881059035946788918972354560, −2.21759540581430457238434984357, −2.07070886115653362971446260255, −0.896464788541896387308152225520, 0.896464788541896387308152225520, 2.07070886115653362971446260255, 2.21759540581430457238434984357, 2.73881059035946788918972354560, 3.10741552901677517301310799421, 3.95239635753751879776209999070, 4.24042279354031696152018996599, 4.89250490777436438282645973149, 5.18283535932327000913724474644, 5.29213107875301605217683294288, 6.19271244943421759707244801744, 6.55378443777929638949203500807, 7.12753286825809124388889063578, 7.46381363676700004773407029293, 7.75157693361575687240855413046, 8.084263798150192620438116216965, 8.381170347648151226574497511031, 8.933022611431709029939379201112, 9.361676175737549223754440137754, 9.613009786047755710339410458238

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