↑

Properties

Label 4-2178e2-1.1-c1e2-0-4
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s + 16-s − 20-s − 2·23-s − 7·25-s + 27-s − 3·31-s − 36-s − 2·37-s + 45-s − 11·47-s + 48-s − 6·49-s − 6·53-s + 3·59-s − 60-s − 64-s − 3·67-s − 2·69-s + 8·71-s − 7·75-s + 80-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 0.447·5-s + 1/3·9-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.223·20-s − 0.417·23-s − 7/5·25-s + 0.192·27-s − 0.538·31-s − 1/6·36-s − 0.328·37-s + 0.149·45-s − 1.60·47-s + 0.144·48-s − 6/7·49-s − 0.824·53-s + 0.390·59-s − 0.129·60-s − 1/8·64-s − 0.366·67-s − 0.240·69-s + 0.949·71-s − 0.808·75-s + 0.111·80-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765499553\)
\(L(\frac12)\) \(\approx\) \(1.765499553\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 - T \)
11 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.ab_i
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.13.a_at
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.17.a_aw
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.19.a_m
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.c_bm
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.d_bs
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.c_bn
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.41.a_ba
43$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.43.a_abk
47$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.l_es
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.g_bi
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.ad_k
61$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.61.a_abh
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.d_fe
71$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.71.ai_cz
73$C_2^2$ \( 1 + 45 T^{2} + p^{2} T^{4} \) 2.73.a_bt
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 + 131 T^{2} + p^{2} T^{4} \) 2.83.a_fb
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.89.ar_hm
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.i_co
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.47682704379833297502951064014, −6.90483643866270621916879904368, −6.44517846909703847696542809136, −6.25445919260678435610900941150, −5.62593486572395552461607446144, −5.26820129412220669873510783345, −4.89688043453987299664035597346, −4.36787297633981588396478239727, −3.84354968595224400074033165544, −3.60391410576657566822697286389, −2.99632285334668156286003677811, −2.49237125251684844700190709711, −1.73646318917312070352661479131, −1.60462027878244259210569888129, −0.43564453505219936092566746195, 0.43564453505219936092566746195, 1.60462027878244259210569888129, 1.73646318917312070352661479131, 2.49237125251684844700190709711, 2.99632285334668156286003677811, 3.60391410576657566822697286389, 3.84354968595224400074033165544, 4.36787297633981588396478239727, 4.89688043453987299664035597346, 5.26820129412220669873510783345, 5.62593486572395552461607446144, 6.25445919260678435610900941150, 6.44517846909703847696542809136, 6.90483643866270621916879904368, 7.47682704379833297502951064014

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