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Properties

Label 4-2178e2-1.1-c1e2-0-16
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 + 12·17-s − 8·25-s + 6·32-s + 24·34-s + 12·37-s + 6·49-s − 16·50-s + 7·64-s − 24·67-s + 36·68-s + 24·74-s − 24·83-s + 20·97-s + 12·98-s − 24·100-s + 8·103-s + 127-s + 8·128-s + 131-s − 48·134-s + 48·136-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s + 2.91·17-s − 8/5·25-s + 1.06·32-s + 4.11·34-s + 1.97·37-s + 6/7·49-s − 2.26·50-s + 7/8·64-s − 2.93·67-s + 4.36·68-s + 2.78·74-s − 2.63·83-s + 2.03·97-s + 1.21·98-s − 2.39·100-s + 0.788·103-s + 0.0887·127-s + 0.707·128-s + 0.0873·131-s − 4.14·134-s + 4.11·136-s + 0.0854·137-s + 0.0848·139-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

\(L(1)\) \(\approx\) \(7.616481470\)
\(L(\frac12)\) \(\approx\) \(7.616481470\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.13.a_ay
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
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$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.43.a_acc
47$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.47.a_aw
53$C_2^2$ \( 1 - 88 T^{2} + p^{2} T^{4} \) 2.53.a_adk
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \) 2.61.a_aeq
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.71.a_acs
73$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \) 2.73.a_ds
79$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.79.a_bq
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 - 160 T^{2} + p^{2} T^{4} \) 2.89.a_age
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.700200193648972818391184791745, −8.839332495907355363986575409224, −8.300830374881008598755165375265, −7.87900174168093536038486652313, −7.60183039355284486323473826122, −7.38948711922697816566161799848, −6.94297007772269062955634882716, −6.16370612713282021447650225664, −6.02902169470239885384633014247, −5.61774395075118013416325273960, −5.50525367698163209615306659943, −4.77241015235792485713826823672, −4.40241357404861589527267299103, −3.99952049493026737030712255266, −3.52741934982351622745064198445, −3.03132880803659991588805893111, −2.81697088301000007007527750358, −1.99562551595579132836604768494, −1.47651064277850281396879045407, −0.78884071350329861832012015590, 0.78884071350329861832012015590, 1.47651064277850281396879045407, 1.99562551595579132836604768494, 2.81697088301000007007527750358, 3.03132880803659991588805893111, 3.52741934982351622745064198445, 3.99952049493026737030712255266, 4.40241357404861589527267299103, 4.77241015235792485713826823672, 5.50525367698163209615306659943, 5.61774395075118013416325273960, 6.02902169470239885384633014247, 6.16370612713282021447650225664, 6.94297007772269062955634882716, 7.38948711922697816566161799848, 7.60183039355284486323473826122, 7.87900174168093536038486652313, 8.300830374881008598755165375265, 8.839332495907355363986575409224, 9.700200193648972818391184791745

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