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Properties

Label 4-2240e2-1.1-c1e2-0-2
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 5·9-s − 2·11-s + 8·19-s + 11·25-s − 2·29-s + 12·31-s − 20·41-s − 20·45-s − 49-s + 8·55-s − 12·59-s + 8·61-s + 32·71-s − 22·79-s + 16·81-s − 24·89-s − 32·95-s − 10·99-s − 30·109-s − 19·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  − 1.78·5-s + 5/3·9-s − 0.603·11-s + 1.83·19-s + 11/5·25-s − 0.371·29-s + 2.15·31-s − 3.12·41-s − 2.98·45-s − 1/7·49-s + 1.07·55-s − 1.56·59-s + 1.02·61-s + 3.79·71-s − 2.47·79-s + 16/9·81-s − 2.54·89-s − 3.28·95-s − 1.00·99-s − 2.87·109-s − 1.72·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.556840375\)
\(L(\frac12)\) \(\approx\) \(1.556840375\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.13.a_az
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.23.a_abq
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.47.a_an
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.71.abg_pi
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2^2$ \( 1 + 167 T^{2} + p^{2} T^{4} \) 2.97.a_gl
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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

−9.420317614107810252822301409860, −8.542424547030526254761347178488, −8.393090239070038574813464941167, −8.025460007028469154778905167366, −7.76250499372321739481278893558, −7.24417849806341420839282060698, −7.02426268440400503282985943107, −6.73101850042924224945473979757, −6.31319940054283498488125816837, −5.46269164017415978342635047579, −5.07564597879093575582859771829, −4.88436179121503404212263908218, −4.35083301824026702807948461899, −3.84390421675141423186622724740, −3.68700826587071253205425586498, −2.93048892982389089818352807581, −2.79199601537814098102097314574, −1.65999599518093312425907452823, −1.24113238503267741471942582820, −0.48688561812222967842279542421, 0.48688561812222967842279542421, 1.24113238503267741471942582820, 1.65999599518093312425907452823, 2.79199601537814098102097314574, 2.93048892982389089818352807581, 3.68700826587071253205425586498, 3.84390421675141423186622724740, 4.35083301824026702807948461899, 4.88436179121503404212263908218, 5.07564597879093575582859771829, 5.46269164017415978342635047579, 6.31319940054283498488125816837, 6.73101850042924224945473979757, 7.02426268440400503282985943107, 7.24417849806341420839282060698, 7.76250499372321739481278893558, 8.025460007028469154778905167366, 8.393090239070038574813464941167, 8.542424547030526254761347178488, 9.420317614107810252822301409860

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