Properties

Label 4-490e2-1.1-c1e2-0-8
Degree $4$
Conductor $240100$
Sign $1$
Analytic cond. $15.3089$
Root an. cond. $1.97804$
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 − 4·5-s + 6·9-s + 6·11-s + 16-s + 10·19-s + 4·20-s + 11·25-s + 8·29-s − 4·31-s − 6·36-s + 6·41-s − 6·44-s − 24·45-s − 24·55-s + 8·59-s + 12·61-s − 64-s − 12·71-s − 10·76-s − 28·79-s − 4·80-s + 27·81-s − 4·89-s − 40·95-s + 36·99-s − 11·100-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s + 2·9-s + 1.80·11-s + 1/4·16-s + 2.29·19-s + 0.894·20-s + 11/5·25-s + 1.48·29-s − 0.718·31-s − 36-s + 0.937·41-s − 0.904·44-s − 3.57·45-s − 3.23·55-s + 1.04·59-s + 1.53·61-s − 1/8·64-s − 1.42·71-s − 1.14·76-s − 3.15·79-s − 0.447·80-s + 3·81-s − 0.423·89-s − 4.10·95-s + 3.61·99-s − 1.09·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15.3089\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 240100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.664971458\)
\(L(\frac12)\) \(\approx\) \(1.664971458\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−11.48947733962775198678491193185, −10.69100982535085619094236129805, −10.23195487658469570204656855462, −9.858366224944203133113634331727, −9.282546635573790288855844113001, −9.137085089351067203624728961973, −8.372242676560961015373059379618, −8.088437259162251802541409190148, −7.31931761305055367988344293878, −7.18106508689761536870910180359, −6.95227880076570226611336555267, −6.19760980646821566432054851269, −5.41580486973594427293227040744, −4.73102116268163805107121819294, −4.38608647121429650792201448110, −3.84495003997242296064139441080, −3.66343780854420297138124817502, −2.84863856857504638710452271731, −1.32450061326598305781130059527, −1.01170395624524908976963510299, 1.01170395624524908976963510299, 1.32450061326598305781130059527, 2.84863856857504638710452271731, 3.66343780854420297138124817502, 3.84495003997242296064139441080, 4.38608647121429650792201448110, 4.73102116268163805107121819294, 5.41580486973594427293227040744, 6.19760980646821566432054851269, 6.95227880076570226611336555267, 7.18106508689761536870910180359, 7.31931761305055367988344293878, 8.088437259162251802541409190148, 8.372242676560961015373059379618, 9.137085089351067203624728961973, 9.282546635573790288855844113001, 9.858366224944203133113634331727, 10.23195487658469570204656855462, 10.69100982535085619094236129805, 11.48947733962775198678491193185

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