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Properties

Label 4-2240e2-1.1-c1e2-0-9
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 6·9-s + 8·19-s − 25-s + 4·29-s − 16·31-s + 12·41-s − 12·45-s − 49-s − 8·59-s + 12·61-s + 24·71-s + 8·79-s + 27·81-s + 20·89-s − 16·95-s + 36·101-s + 28·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s + 2·9-s + 1.83·19-s − 1/5·25-s + 0.742·29-s − 2.87·31-s + 1.87·41-s − 1.78·45-s − 1/7·49-s − 1.04·59-s + 1.53·61-s + 2.84·71-s + 0.900·79-s + 3·81-s + 2.11·89-s − 1.64·95-s + 3.58·101-s + 2.68·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-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

\(L(1)\) \(\approx\) \(2.840423748\)
\(L(\frac12)\) \(\approx\) \(2.840423748\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−9.153473697376734019700972831067, −9.137686707900680085913476179534, −8.314695529438921143449331470373, −7.85736555584818748408962285992, −7.67548951854078018943757900977, −7.36280658329113588010497697489, −7.01640001503473046161691473715, −6.75136866520401204692980621471, −6.01796098683231584210676222744, −5.76237766064378023577312360636, −5.06992073339014491621673593167, −4.83554825728377376280551939183, −4.41892627409401099006273881689, −3.80896328457004026173143164995, −3.48217703859000083218707859499, −3.38750467354747029544868907039, −2.16125677535040774435472701818, −2.05657384643093579597724417832, −1.09118947008733293021346928441, −0.70871702573805387919943549440, 0.70871702573805387919943549440, 1.09118947008733293021346928441, 2.05657384643093579597724417832, 2.16125677535040774435472701818, 3.38750467354747029544868907039, 3.48217703859000083218707859499, 3.80896328457004026173143164995, 4.41892627409401099006273881689, 4.83554825728377376280551939183, 5.06992073339014491621673593167, 5.76237766064378023577312360636, 6.01796098683231584210676222744, 6.75136866520401204692980621471, 7.01640001503473046161691473715, 7.36280658329113588010497697489, 7.67548951854078018943757900977, 7.85736555584818748408962285992, 8.314695529438921143449331470373, 9.137686707900680085913476179534, 9.153473697376734019700972831067

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