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Properties

Label 4-990e2-1.1-c1e2-0-43
Degree $4$
Conductor $980100$
Sign $1$
Analytic cond. $62.4920$
Root an. cond. $2.81161$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·5-s + 7-s + 4·8-s − 4·10-s + 2·11-s + 4·13-s + 2·14-s + 5·16-s + 3·17-s + 7·19-s − 6·20-s + 4·22-s + 6·23-s + 3·25-s + 8·26-s + 3·28-s + 3·29-s + 31-s + 6·32-s + 6·34-s − 2·35-s + 13·37-s + 14·38-s − 8·40-s − 8·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s + 1.41·8-s − 1.26·10-s + 0.603·11-s + 1.10·13-s + 0.534·14-s + 5/4·16-s + 0.727·17-s + 1.60·19-s − 1.34·20-s + 0.852·22-s + 1.25·23-s + 3/5·25-s + 1.56·26-s + 0.566·28-s + 0.557·29-s + 0.179·31-s + 1.06·32-s + 1.02·34-s − 0.338·35-s + 2.13·37-s + 2.27·38-s − 1.26·40-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(980100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(62.4920\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 980100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.289079855\)
\(L(\frac12)\) \(\approx\) \(6.289079855\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$D_{4}$ \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_bc
19$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.19.ah_bq
23$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_w
29$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_ca
31$D_{4}$ \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_cc
37$D_{4}$ \( 1 - 13 T + 108 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.37.an_ee
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cs
53$D_{4}$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_ca
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_dq
61$D_{4}$ \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_eq
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$D_{4}$ \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_cs
73$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_be
79$D_{4}$ \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_gs
83$D_{4}$ \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_fm
89$D_{4}$ \( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.89.d_gq
97$D_{4}$ \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_ic
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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

−10.18117660127891089939227519679, −9.987167497693696907981490082249, −9.274576879196523008943501056100, −9.033087181020552502331228861761, −8.215677478566040724031925613344, −8.122791105403659904684444900838, −7.55886488555460606639278772860, −7.24890440236121834873346396090, −6.59502373629883656343250751602, −6.45968307937970757559734997664, −5.59914590068054735675065520320, −5.58519302697905327488577221262, −4.75387261308194036035165751564, −4.54878270126902218209876285934, −3.98407204612482346710023372520, −3.42710888140596097956805943737, −3.10831468308433265340113329022, −2.63434882131933127885518227886, −1.34411130567678245509630805146, −1.16615174006350466134066828480, 1.16615174006350466134066828480, 1.34411130567678245509630805146, 2.63434882131933127885518227886, 3.10831468308433265340113329022, 3.42710888140596097956805943737, 3.98407204612482346710023372520, 4.54878270126902218209876285934, 4.75387261308194036035165751564, 5.58519302697905327488577221262, 5.59914590068054735675065520320, 6.45968307937970757559734997664, 6.59502373629883656343250751602, 7.24890440236121834873346396090, 7.55886488555460606639278772860, 8.122791105403659904684444900838, 8.215677478566040724031925613344, 9.033087181020552502331228861761, 9.274576879196523008943501056100, 9.987167497693696907981490082249, 10.18117660127891089939227519679

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