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Properties

Label 4-220e2-1.1-c1e2-0-6
Degree $4$
Conductor $48400$
Sign $1$
Analytic cond. $3.08602$
Root an. cond. $1.32540$
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  − 2·4-s − 2·5-s + 8·7-s + 4·9-s + 6·11-s + 4·16-s − 4·19-s + 4·20-s + 3·25-s − 16·28-s − 16·35-s − 8·36-s − 20·37-s − 16·43-s − 12·44-s − 8·45-s + 34·49-s + 12·53-s − 12·55-s + 32·63-s − 8·64-s + 8·76-s + 48·77-s − 4·79-s − 8·80-s + 7·81-s − 24·83-s + ⋯
L(s)  = 1  − 4-s − 0.894·5-s + 3.02·7-s + 4/3·9-s + 1.80·11-s + 16-s − 0.917·19-s + 0.894·20-s + 3/5·25-s − 3.02·28-s − 2.70·35-s − 4/3·36-s − 3.28·37-s − 2.43·43-s − 1.80·44-s − 1.19·45-s + 34/7·49-s + 1.64·53-s − 1.61·55-s + 4.03·63-s − 64-s + 0.917·76-s + 5.47·77-s − 0.450·79-s − 0.894·80-s + 7/9·81-s − 2.63·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(48400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.08602\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 48400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.538348007\)
\(L(\frac12)\) \(\approx\) \(1.538348007\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.17.a_abg
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.23.a_e
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.41.a_acw
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.47.a_abs
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.59.a_aeg
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.61.a_aby
67$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \) 2.67.a_aem
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.71.a_aeg
73$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \) 2.73.a_aey
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−12.32572159225173888534868318232, −11.91934166331702961527607756285, −11.69713197133463721218562939558, −11.23053615138914854274494805200, −10.42720885594776132092103681591, −10.41513279393201236750496220607, −9.544158882161330291819850891124, −8.767396873938303140236528706261, −8.461249364944985126535333423822, −8.387923182954024130896192709445, −7.57401679828664294672075292699, −7.09540129258616420149287833856, −6.65964606940817689731672563890, −5.42541883623770981186067896708, −5.01451777206188100954596589382, −4.46632048790958852214627371168, −4.10392218391271091879316423678, −3.63542823351505933271264626737, −1.65409330999915474272984156560, −1.47376018620487891796086728364, 1.47376018620487891796086728364, 1.65409330999915474272984156560, 3.63542823351505933271264626737, 4.10392218391271091879316423678, 4.46632048790958852214627371168, 5.01451777206188100954596589382, 5.42541883623770981186067896708, 6.65964606940817689731672563890, 7.09540129258616420149287833856, 7.57401679828664294672075292699, 8.387923182954024130896192709445, 8.461249364944985126535333423822, 8.767396873938303140236528706261, 9.544158882161330291819850891124, 10.41513279393201236750496220607, 10.42720885594776132092103681591, 11.23053615138914854274494805200, 11.69713197133463721218562939558, 11.91934166331702961527607756285, 12.32572159225173888534868318232

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