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Properties

Label 4-2178e2-1.1-c1e2-0-12
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 8·17-s + 2·25-s + 20·29-s − 6·32-s − 16·34-s + 12·37-s − 4·49-s − 4·50-s − 40·58-s + 7·64-s + 16·67-s + 24·68-s − 24·74-s − 16·83-s + 8·98-s + 6·100-s − 20·101-s + 28·103-s + 60·116-s + 127-s − 8·128-s + 131-s − 32·134-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 5/4·16-s + 1.94·17-s + 2/5·25-s + 3.71·29-s − 1.06·32-s − 2.74·34-s + 1.97·37-s − 4/7·49-s − 0.565·50-s − 5.25·58-s + 7/8·64-s + 1.95·67-s + 2.91·68-s − 2.78·74-s − 1.75·83-s + 0.808·98-s + 3/5·100-s − 1.99·101-s + 2.75·103-s + 5.57·116-s + 0.0887·127-s − 0.707·128-s + 0.0873·131-s − 2.76·134-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.687179809\)
\(L(\frac12)\) \(\approx\) \(1.687179809\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
11 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.23.a_abs
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.29.au_gc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.47.a_ado
53$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.53.a_adu
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.61.a_aby
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.71.a_u
73$C_2^2$ \( 1 - 144 T^{2} + p^{2} T^{4} \) 2.73.a_afo
79$C_2^2$ \( 1 - 108 T^{2} + p^{2} T^{4} \) 2.79.a_aee
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \) 2.89.a_ge
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−9.464720148285383274395083211133, −8.600506626386974754733153599815, −8.591719854503404647973526606440, −8.060706767888440493666686375696, −7.994022171981303662318707903022, −7.42427370593395791912102636538, −7.03432867991204954792087472939, −6.61932953362489900343342617688, −6.28591626832989856474494922376, −5.86058063485458486945352480239, −5.45735686938429888370837138293, −4.79266877933545202269196340713, −4.56586817801460336750958268308, −3.81068719113730444364575991074, −3.25212140046988975363211780834, −2.76071446101906306769807303111, −2.56905041531577077988028504277, −1.64879463858937532460343046813, −0.981928877622687933044407594463, −0.76326666849079354093271603211, 0.76326666849079354093271603211, 0.981928877622687933044407594463, 1.64879463858937532460343046813, 2.56905041531577077988028504277, 2.76071446101906306769807303111, 3.25212140046988975363211780834, 3.81068719113730444364575991074, 4.56586817801460336750958268308, 4.79266877933545202269196340713, 5.45735686938429888370837138293, 5.86058063485458486945352480239, 6.28591626832989856474494922376, 6.61932953362489900343342617688, 7.03432867991204954792087472939, 7.42427370593395791912102636538, 7.994022171981303662318707903022, 8.060706767888440493666686375696, 8.591719854503404647973526606440, 8.600506626386974754733153599815, 9.464720148285383274395083211133

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