Properties

Label 4-6650e2-1.1-c1e2-0-3
Degree $4$
Conductor $44222500$
Sign $1$
Analytic cond. $2819.66$
Root an. cond. $7.28701$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 2·7-s − 4·8-s − 6·12-s + 2·13-s + 4·14-s + 5·16-s − 12·17-s + 2·19-s + 4·21-s + 8·24-s − 4·26-s + 2·27-s − 6·28-s + 12·29-s − 2·31-s − 6·32-s + 24·34-s + 2·37-s − 4·38-s − 4·39-s − 6·41-s − 8·42-s − 10·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s − 1.41·8-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 5/4·16-s − 2.91·17-s + 0.458·19-s + 0.872·21-s + 1.63·24-s − 0.784·26-s + 0.384·27-s − 1.13·28-s + 2.22·29-s − 0.359·31-s − 1.06·32-s + 4.11·34-s + 0.328·37-s − 0.648·38-s − 0.640·39-s − 0.937·41-s − 1.23·42-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44222500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2819.66\)
Root analytic conductor: \(7.28701\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44222500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5605315998\)
\(L(\frac12)\) \(\approx\) \(0.5605315998\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
11$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.11.a_t
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$D_{4}$ \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cp
23$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.23.a_abd
29$D_{4}$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_dn
31$D_{4}$ \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_bk
37$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_bw
41$D_{4}$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_br
43$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dg
47$D_{4}$ \( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.47.as_gq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \) 2.59.a_el
61$D_{4}$ \( 1 - 4 T + 99 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_dv
67$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_bb
71$D_{4}$ \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_dz
73$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_cc
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.79.aw_kt
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.83.g_gt
89$D_{4}$ \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hu
97$D_{4}$ \( 1 - 2 T + 168 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_gm
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

−8.275752991057001402540928932383, −7.909648892156576576815748759453, −7.42944515398781858006261904910, −6.94067646743833235727622717448, −6.72477942928488392955150402326, −6.56597261686210845549812497519, −6.15873008237211290173407854881, −6.03585075183883781093507459163, −5.27735730083693390664472884423, −5.23345353961057386743869976393, −4.60779009648499545217231981536, −4.28754341272158613279342036326, −3.65677307517452799732689964877, −3.40241974419671381644488881920, −2.57389574324208106182683349958, −2.53977852052802197084772245671, −1.99697875223659112087411396095, −1.35174073242874411175114074398, −0.56687921305132432990296558432, −0.47728002895098514431562076337, 0.47728002895098514431562076337, 0.56687921305132432990296558432, 1.35174073242874411175114074398, 1.99697875223659112087411396095, 2.53977852052802197084772245671, 2.57389574324208106182683349958, 3.40241974419671381644488881920, 3.65677307517452799732689964877, 4.28754341272158613279342036326, 4.60779009648499545217231981536, 5.23345353961057386743869976393, 5.27735730083693390664472884423, 6.03585075183883781093507459163, 6.15873008237211290173407854881, 6.56597261686210845549812497519, 6.72477942928488392955150402326, 6.94067646743833235727622717448, 7.42944515398781858006261904910, 7.909648892156576576815748759453, 8.275752991057001402540928932383

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