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Properties

Label 4-650e2-1.1-c1e2-0-1
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  − 4-s − 4·7-s + 2·11-s − 6·13-s + 16-s − 4·17-s − 6·19-s − 2·23-s + 4·28-s + 12·31-s − 4·37-s + 2·41-s − 12·43-s − 2·44-s + 16·47-s − 2·49-s + 6·52-s − 10·53-s + 6·59-s + 4·61-s − 64-s + 4·68-s − 8·71-s + 6·76-s − 8·77-s − 9·81-s + 8·83-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.51·7-s + 0.603·11-s − 1.66·13-s + 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.417·23-s + 0.755·28-s + 2.15·31-s − 0.657·37-s + 0.312·41-s − 1.82·43-s − 0.301·44-s + 2.33·47-s − 2/7·49-s + 0.832·52-s − 1.37·53-s + 0.781·59-s + 0.512·61-s − 1/8·64-s + 0.485·68-s − 0.949·71-s + 0.688·76-s − 0.911·77-s − 81-s + 0.878·83-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

\(L(1)\) \(\approx\) \(0.5011602663\)
\(L(\frac12)\) \(\approx\) \(0.5011602663\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_c
17$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_i
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_s
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_cu
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ac_c
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_cu
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.k_by
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_s
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_bg
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_du
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−10.50216950673101279494828242424, −10.19836000906053671712503157702, −9.880701168651538483545524427534, −9.626441914554065124511204746133, −9.034187051186593356099017083714, −8.631625502707661486784228531783, −8.384337284992463405107913412477, −7.61322566512906037076969112060, −7.20018038638038360090307323925, −6.66625017792408505921308474131, −6.24644591830586329219217853269, −6.16080015470419843055350496955, −5.05141089402001287832938869078, −4.88726671243182486822197441193, −4.11317476822675674440714923185, −3.90163357213345578701726506416, −2.92181579019591234945792961357, −2.64077184553483477963161860839, −1.78509481947871277327536463665, −0.36989660787846947140103602225, 0.36989660787846947140103602225, 1.78509481947871277327536463665, 2.64077184553483477963161860839, 2.92181579019591234945792961357, 3.90163357213345578701726506416, 4.11317476822675674440714923185, 4.88726671243182486822197441193, 5.05141089402001287832938869078, 6.16080015470419843055350496955, 6.24644591830586329219217853269, 6.66625017792408505921308474131, 7.20018038638038360090307323925, 7.61322566512906037076969112060, 8.384337284992463405107913412477, 8.631625502707661486784228531783, 9.034187051186593356099017083714, 9.626441914554065124511204746133, 9.880701168651538483545524427534, 10.19836000906053671712503157702, 10.50216950673101279494828242424

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