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Properties

Label 4-210e2-1.1-c1e2-0-1
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $2.81185$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s − 2·5-s + 4·7-s + 9-s + 2·12-s + 4·15-s + 16-s + 2·20-s − 8·21-s − 25-s + 4·27-s − 4·28-s − 8·35-s − 36-s − 8·37-s + 16·41-s − 2·45-s + 8·47-s − 2·48-s + 9·49-s − 8·59-s − 4·60-s + 4·63-s − 64-s + 2·75-s + 24·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.577·12-s + 1.03·15-s + 1/4·16-s + 0.447·20-s − 1.74·21-s − 1/5·25-s + 0.769·27-s − 0.755·28-s − 1.35·35-s − 1/6·36-s − 1.31·37-s + 2.49·41-s − 0.298·45-s + 1.16·47-s − 0.288·48-s + 9/7·49-s − 1.04·59-s − 0.516·60-s + 0.503·63-s − 1/8·64-s + 0.230·75-s + 2.70·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.81185\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7101432331\)
\(L(\frac12)\) \(\approx\) \(0.7101432331\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.i_cc
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.41.aq_fa
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.i_bi
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.73.a_s
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.79.ay_lq
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.e_ec
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.aq_gw
97$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.97.a_de
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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

−10.54433870099029857190801101464, −9.706236058192782324119483304271, −9.085851832629691146797632008844, −8.598431184881358711703949196792, −8.077128682028555096600755801352, −7.55697688838653064107522343938, −7.22721543427899875189086913032, −6.26017261888074639741861700209, −5.79894702636093395214072050177, −5.15102041134261357260933885778, −4.67658093059294008723111999506, −4.21091930040045726187959489307, −3.41274275754831116425270873938, −2.13373457280465465845736740479, −0.821152998784911266442281957803, 0.821152998784911266442281957803, 2.13373457280465465845736740479, 3.41274275754831116425270873938, 4.21091930040045726187959489307, 4.67658093059294008723111999506, 5.15102041134261357260933885778, 5.79894702636093395214072050177, 6.26017261888074639741861700209, 7.22721543427899875189086913032, 7.55697688838653064107522343938, 8.077128682028555096600755801352, 8.598431184881358711703949196792, 9.085851832629691146797632008844, 9.706236058192782324119483304271, 10.54433870099029857190801101464

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