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Properties

Label 4-650e2-1.1-c1e2-0-12
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-s − 3·7-s − 8-s + 3·9-s + 3·11-s + 7·13-s − 3·14-s − 16-s − 4·17-s + 3·18-s − 7·19-s + 3·22-s − 4·23-s + 7·26-s + 8·29-s − 20·31-s − 4·34-s + 3·37-s − 7·38-s + 2·41-s + 6·43-s − 4·46-s + 2·47-s + 7·49-s + 18·53-s + 3·56-s + 8·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.13·7-s − 0.353·8-s + 9-s + 0.904·11-s + 1.94·13-s − 0.801·14-s − 1/4·16-s − 0.970·17-s + 0.707·18-s − 1.60·19-s + 0.639·22-s − 0.834·23-s + 1.37·26-s + 1.48·29-s − 3.59·31-s − 0.685·34-s + 0.493·37-s − 1.13·38-s + 0.312·41-s + 0.914·43-s − 0.589·46-s + 0.291·47-s + 49-s + 2.47·53-s + 0.400·56-s + 1.05·58-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

\(L(1)\) \(\approx\) \(2.389548102\)
\(L(\frac12)\) \(\approx\) \(2.389548102\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_bj
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_abc
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ah
47$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.47.ac_dr
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abr
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.61.ao_ff
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2^2$ \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_gd
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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

−10.79017168823907123621988358100, −10.56137925077447357653295631519, −9.812177129001117452654268639964, −9.518812864310649863309719700539, −8.913744528232178928071515871359, −8.721345803051156553388781856967, −8.401474518658479747614416471365, −7.52647273559935246481970762978, −6.99717128473903798836486865976, −6.65890804551353969216972077966, −6.36433241489654664088541713359, −5.68084442415512962177400840998, −5.62798750014615608750309659937, −4.39678740192933966420872304385, −4.23378446391115973481511762787, −3.69638476857268843424200575725, −3.55952613992839848789754558056, −2.39196469143154050840628260786, −1.88264824026137694226345291465, −0.77054046507760470462636308043, 0.77054046507760470462636308043, 1.88264824026137694226345291465, 2.39196469143154050840628260786, 3.55952613992839848789754558056, 3.69638476857268843424200575725, 4.23378446391115973481511762787, 4.39678740192933966420872304385, 5.62798750014615608750309659937, 5.68084442415512962177400840998, 6.36433241489654664088541713359, 6.65890804551353969216972077966, 6.99717128473903798836486865976, 7.52647273559935246481970762978, 8.401474518658479747614416471365, 8.721345803051156553388781856967, 8.913744528232178928071515871359, 9.518812864310649863309719700539, 9.812177129001117452654268639964, 10.56137925077447357653295631519, 10.79017168823907123621988358100

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