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Properties

Label 4-7105e2-1.1-c1e2-0-3
Degree $4$
Conductor $50481025$
Sign $1$
Analytic cond. $3218.71$
Root an. cond. $7.53217$
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  − 4-s + 2·5-s − 6·9-s − 4·11-s + 4·13-s − 3·16-s − 4·19-s − 2·20-s + 3·25-s + 2·29-s + 4·31-s + 6·36-s − 4·41-s + 4·44-s − 12·45-s + 8·47-s − 4·52-s − 4·53-s − 8·55-s + 16·59-s − 4·61-s + 7·64-s + 8·65-s − 8·67-s − 16·71-s − 8·73-s + 4·76-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s − 2·9-s − 1.20·11-s + 1.10·13-s − 3/4·16-s − 0.917·19-s − 0.447·20-s + 3/5·25-s + 0.371·29-s + 0.718·31-s + 36-s − 0.624·41-s + 0.603·44-s − 1.78·45-s + 1.16·47-s − 0.554·52-s − 0.549·53-s − 1.07·55-s + 2.08·59-s − 0.512·61-s + 7/8·64-s + 0.992·65-s − 0.977·67-s − 1.89·71-s − 0.936·73-s + 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50481025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50481025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50481025\)    =    \(5^{2} \cdot 7^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(3218.71\)
Root analytic conductor: \(7.53217\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50481025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.040285068\)
\(L(\frac12)\) \(\approx\) \(2.040285068\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
31$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_s
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cw
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$D_{4}$ \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_ck
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_ek
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_cc
79$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_fu
83$D_{4}$ \( 1 - 16 T + 182 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_ha
89$D_{4}$ \( 1 - 28 T + 362 T^{2} - 28 p T^{3} + p^{2} T^{4} \) 2.89.abc_ny
97$D_{4}$ \( 1 - 16 T + 246 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_jm
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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

−8.091893689260667357912453835623, −7.913399817769832237921042076028, −7.50866116450495643441454210584, −7.05712100957669097187161233363, −6.47717323364258879772992099514, −6.16001574631107038626418748651, −6.12780661157302375599343746096, −5.75601590479250731972507535315, −5.19114364622544366948955309729, −5.07515500916270937368533768754, −4.60781403894123865048649320990, −4.32943011806704704915471897479, −3.49752688860009174779579485625, −3.42482563955295007436368372236, −2.91875126090396573346438931567, −2.31064101302835281633468553459, −2.29704650583877633179173676940, −1.71991611475297263492075794104, −0.70810508085074477967493143344, −0.49615108747465581667819550554, 0.49615108747465581667819550554, 0.70810508085074477967493143344, 1.71991611475297263492075794104, 2.29704650583877633179173676940, 2.31064101302835281633468553459, 2.91875126090396573346438931567, 3.42482563955295007436368372236, 3.49752688860009174779579485625, 4.32943011806704704915471897479, 4.60781403894123865048649320990, 5.07515500916270937368533768754, 5.19114364622544366948955309729, 5.75601590479250731972507535315, 6.12780661157302375599343746096, 6.16001574631107038626418748651, 6.47717323364258879772992099514, 7.05712100957669097187161233363, 7.50866116450495643441454210584, 7.913399817769832237921042076028, 8.091893689260667357912453835623

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