Properties

Label 4-1014e2-1.1-c1e2-0-11
Degree $4$
Conductor $1028196$
Sign $1$
Analytic cond. $65.5586$
Root an. cond. $2.84549$
Motivic weight $1$
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  + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s + 6·17-s + 12·23-s + 25-s + 4·27-s + 6·29-s − 3·36-s + 20·43-s + 2·48-s + 10·49-s + 12·51-s + 6·53-s − 14·61-s − 64-s − 6·68-s + 24·69-s + 2·75-s − 8·79-s + 5·81-s + 12·87-s − 12·92-s − 100-s − 30·101-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s + 1.45·17-s + 2.50·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 1/2·36-s + 3.04·43-s + 0.288·48-s + 10/7·49-s + 1.68·51-s + 0.824·53-s − 1.79·61-s − 1/8·64-s − 0.727·68-s + 2.88·69-s + 0.230·75-s − 0.900·79-s + 5/9·81-s + 1.28·87-s − 1.25·92-s − 0.0999·100-s − 2.98·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1028196\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(65.5586\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1014} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1028196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.664237604\)
\(L(\frac12)\) \(\approx\) \(3.664237604\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−10.09694217243966762563634774920, −9.498289266918123704869803459900, −9.255335132039047088910702636810, −9.078722535519452308386621994420, −8.491109486648388807856001787039, −8.219978263515380867421639487113, −7.62238022081785491445680933340, −7.42479235173276700978165342461, −6.92275062589676140230407220281, −6.51585432110654090064465268534, −5.62108289814491207044824502621, −5.52830597670793300481902384933, −4.88452264775728862718748791489, −4.23834804277256824000914338209, −4.04204691639570562154107595401, −3.27399901836417283259677689636, −2.70202374681723769908966333305, −2.66465392061265434602738065832, −1.30260739923111111729756136444, −1.00201846058436211373320105939, 1.00201846058436211373320105939, 1.30260739923111111729756136444, 2.66465392061265434602738065832, 2.70202374681723769908966333305, 3.27399901836417283259677689636, 4.04204691639570562154107595401, 4.23834804277256824000914338209, 4.88452264775728862718748791489, 5.52830597670793300481902384933, 5.62108289814491207044824502621, 6.51585432110654090064465268534, 6.92275062589676140230407220281, 7.42479235173276700978165342461, 7.62238022081785491445680933340, 8.219978263515380867421639487113, 8.491109486648388807856001787039, 9.078722535519452308386621994420, 9.255335132039047088910702636810, 9.498289266918123704869803459900, 10.09694217243966762563634774920

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