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Properties

Label 4-4650e2-1.1-c1e2-0-8
Degree $4$
Conductor $21622500$
Sign $1$
Analytic cond. $1378.66$
Root an. cond. $6.09347$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 7-s − 4·8-s + 3·9-s + 5·11-s − 6·12-s − 12·13-s − 2·14-s + 5·16-s + 8·17-s − 6·18-s + 19-s − 2·21-s − 10·22-s − 5·23-s + 8·24-s + 24·26-s − 4·27-s + 3·28-s + 2·31-s − 6·32-s − 10·33-s − 16·34-s + 9·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.377·7-s − 1.41·8-s + 9-s + 1.50·11-s − 1.73·12-s − 3.32·13-s − 0.534·14-s + 5/4·16-s + 1.94·17-s − 1.41·18-s + 0.229·19-s − 0.436·21-s − 2.13·22-s − 1.04·23-s + 1.63·24-s + 4.70·26-s − 0.769·27-s + 0.566·28-s + 0.359·31-s − 1.06·32-s − 1.74·33-s − 2.74·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21622500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(1378.66\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21622500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4981466970\)
\(L(\frac12)\) \(\approx\) \(0.4981466970\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_ac
11$D_{4}$ \( 1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_m
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_w
23$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bk
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
37$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_k
41$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_s
43$D_{4}$ \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.43.aj_dm
47$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_bm
53$D_{4}$ \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_do
59$D_{4}$ \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_cc
61$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_co
67$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_da
71$D_{4}$ \( 1 + 17 T + 198 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.71.r_hq
73$D_{4}$ \( 1 - 9 T + 150 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.73.aj_fu
79$D_{4}$ \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.79.j_gg
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$D_{4}$ \( 1 + 5 T + 168 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.89.f_gm
97$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.97.a_aco
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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

−8.268447229500425825361775877490, −8.153934871379927956414994173322, −7.56408231284478756861025936390, −7.53222033303086431425640823331, −7.18024192622616061566564159895, −6.82573054175585324988429718743, −6.39776535727201225040594457088, −6.07501867292663783385547774706, −5.51376186232524458228786901041, −5.40398441213482391157805645112, −4.82941021711112342861154095291, −4.55542202858710814296390487616, −4.01747719748240670356670143382, −3.53793292344824818172434057768, −2.84830918086831138604207225624, −2.57668487827483542201620935882, −1.81305907596016115336129655449, −1.58084702824352474629991976218, −0.916770209767980851110675105888, −0.32954572545734250328408348105, 0.32954572545734250328408348105, 0.916770209767980851110675105888, 1.58084702824352474629991976218, 1.81305907596016115336129655449, 2.57668487827483542201620935882, 2.84830918086831138604207225624, 3.53793292344824818172434057768, 4.01747719748240670356670143382, 4.55542202858710814296390487616, 4.82941021711112342861154095291, 5.40398441213482391157805645112, 5.51376186232524458228786901041, 6.07501867292663783385547774706, 6.39776535727201225040594457088, 6.82573054175585324988429718743, 7.18024192622616061566564159895, 7.53222033303086431425640823331, 7.56408231284478756861025936390, 8.153934871379927956414994173322, 8.268447229500425825361775877490

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