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Properties

Label 4-650e2-1.1-c1e2-0-5
Degree $4$
Conductor $422500$
Sign $1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
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  − 2·2-s + 3·4-s − 4·8-s + 2·9-s − 6·13-s + 5·16-s − 4·18-s + 12·26-s − 12·29-s − 6·32-s + 6·36-s + 12·37-s − 24·47-s − 14·49-s − 18·52-s + 24·58-s + 20·61-s + 7·64-s − 24·67-s − 8·72-s + 12·73-s − 24·74-s + 16·79-s − 5·81-s − 24·83-s + 48·94-s + 36·97-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s − 1.66·13-s + 5/4·16-s − 0.942·18-s + 2.35·26-s − 2.22·29-s − 1.06·32-s + 36-s + 1.97·37-s − 3.50·47-s − 2·49-s − 2.49·52-s + 3.15·58-s + 2.56·61-s + 7/8·64-s − 2.93·67-s − 0.942·72-s + 1.40·73-s − 2.78·74-s + 1.80·79-s − 5/9·81-s − 2.63·83-s + 4.95·94-s + 3.65·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(422500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(26.9389\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6093629831\)
\(L(\frac12)\) \(\approx\) \(0.6093629831\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.97.abk_ty
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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

−10.61936807274272778021164034426, −10.03316412522753416286147133926, −9.876213824945739776814976696835, −9.492341932860479083191995464389, −9.313478676694928712128248966662, −8.564635655145774684766379214295, −8.146344418580692682987618376928, −7.61288131411055002934440299297, −7.53998109098810475670411649059, −6.91307083528489320124079515969, −6.55113378925142431660655389298, −5.98002391130188636884810674640, −5.39712771134260145217386919105, −4.76986491846830856478672218879, −4.36310071733014523680322069996, −3.39019625402270187535838872541, −3.00491013221464840196136923641, −1.94967757998084172562765226889, −1.84841971957976754733369130889, −0.51619564719349802105763267625, 0.51619564719349802105763267625, 1.84841971957976754733369130889, 1.94967757998084172562765226889, 3.00491013221464840196136923641, 3.39019625402270187535838872541, 4.36310071733014523680322069996, 4.76986491846830856478672218879, 5.39712771134260145217386919105, 5.98002391130188636884810674640, 6.55113378925142431660655389298, 6.91307083528489320124079515969, 7.53998109098810475670411649059, 7.61288131411055002934440299297, 8.146344418580692682987618376928, 8.564635655145774684766379214295, 9.313478676694928712128248966662, 9.492341932860479083191995464389, 9.876213824945739776814976696835, 10.03316412522753416286147133926, 10.61936807274272778021164034426

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