↑

Properties

Label 4-4600e2-1.1-c1e2-0-6
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·7-s + 2·9-s + 2·11-s + 4·17-s + 2·19-s − 8·21-s − 2·23-s + 6·27-s − 8·29-s − 8·31-s + 4·33-s + 18·37-s + 6·41-s + 14·43-s + 10·47-s + 3·49-s + 8·51-s − 2·53-s + 4·57-s + 2·59-s − 10·61-s − 8·63-s + 8·67-s − 4·69-s + 20·71-s − 10·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s + 2/3·9-s + 0.603·11-s + 0.970·17-s + 0.458·19-s − 1.74·21-s − 0.417·23-s + 1.15·27-s − 1.48·29-s − 1.43·31-s + 0.696·33-s + 2.95·37-s + 0.937·41-s + 2.13·43-s + 1.45·47-s + 3/7·49-s + 1.12·51-s − 0.274·53-s + 0.529·57-s + 0.260·59-s − 1.28·61-s − 1.00·63-s + 0.977·67-s − 0.481·69-s + 2.37·71-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.323510784\)
\(L(\frac12)\) \(\approx\) \(4.323510784\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_n
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.11.ac_x
13$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.13.a_v
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_s
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
29$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cr
31$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cg
37$D_{4}$ \( 1 - 18 T + 150 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.37.as_fu
41$D_{4}$ \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_l
43$D_{4}$ \( 1 - 14 T + 115 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_el
47$D_{4}$ \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_cw
53$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_dy
59$D_{4}$ \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_ek
61$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_fm
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.71.au_ji
73$D_{4}$ \( 1 + 10 T + 151 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_fv
79$D_{4}$ \( 1 - 8 T + 49 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_bx
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_dq
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

−8.259327125608239487620625767939, −8.245541838200889587305819095577, −7.71283355856066651815143832198, −7.58497734860849402406955210640, −7.08262146568224772221933294188, −6.77848240889571913938302852737, −6.36314772386023370896706148471, −5.89743284607374795219123428212, −5.68465014782805229123921118187, −5.38123284023008561101769664373, −4.59796293739742178973279023073, −4.13654835543416684391157027039, −3.89908739598071179764982796108, −3.59259054468692234951187111224, −2.92887135684917931594154670338, −2.90358027288282905760693584927, −2.30849872191347206761054571249, −1.84145089025482935113419135909, −0.988405524164736964279785325033, −0.62910679321861749837331227631, 0.62910679321861749837331227631, 0.988405524164736964279785325033, 1.84145089025482935113419135909, 2.30849872191347206761054571249, 2.90358027288282905760693584927, 2.92887135684917931594154670338, 3.59259054468692234951187111224, 3.89908739598071179764982796108, 4.13654835543416684391157027039, 4.59796293739742178973279023073, 5.38123284023008561101769664373, 5.68465014782805229123921118187, 5.89743284607374795219123428212, 6.36314772386023370896706148471, 6.77848240889571913938302852737, 7.08262146568224772221933294188, 7.58497734860849402406955210640, 7.71283355856066651815143832198, 8.245541838200889587305819095577, 8.259327125608239487620625767939

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