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Properties

Label 4-210e2-1.1-c1e2-0-9
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  + 3-s − 4-s + 4·5-s − 2·7-s − 2·9-s − 12-s + 4·15-s + 16-s + 9·17-s − 4·20-s − 2·21-s + 11·25-s − 5·27-s + 2·28-s − 8·35-s + 2·36-s + 16·37-s + 41-s − 12·43-s − 8·45-s − 16·47-s + 48-s − 3·49-s + 9·51-s + 10·59-s − 4·60-s + 4·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1.78·5-s − 0.755·7-s − 2/3·9-s − 0.288·12-s + 1.03·15-s + 1/4·16-s + 2.18·17-s − 0.894·20-s − 0.436·21-s + 11/5·25-s − 0.962·27-s + 0.377·28-s − 1.35·35-s + 1/3·36-s + 2.63·37-s + 0.156·41-s − 1.82·43-s − 1.19·45-s − 2.33·47-s + 0.144·48-s − 3/7·49-s + 1.26·51-s + 1.30·59-s − 0.516·60-s + 0.503·63-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

\(L(1)\) \(\approx\) \(1.795911192\)
\(L(\frac12)\) \(\approx\) \(1.795911192\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
good11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
13$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.13.a_f
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.aj_bw
19$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.19.a_x
23$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.23.a_z
29$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.29.a_abl
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.31.a_abl
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ab_cy
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.m_dt
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.53.a_dh
59$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.59.ak_br
61$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.61.a_dz
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.ag_br
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.71.a_abq
73$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \) 2.73.a_aeb
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.n_hu
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.89.ak_dz
97$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.97.a_cs
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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

−9.878178981329807671425407958060, −9.648513446796019982492486605924, −9.515885990727354254532371328665, −8.685097012833614297337989610175, −8.209674498885084252064672125274, −7.79831504823702382483865253714, −6.90437895775839882981654791774, −6.26148239397993775233190421295, −5.90451972362547634326313493976, −5.34204930324312122153102803384, −4.83253515755532556749657600380, −3.63218376032489798597574207201, −3.11371579285855222445428509042, −2.45759284038842526311312554286, −1.33015016148020136712410914502, 1.33015016148020136712410914502, 2.45759284038842526311312554286, 3.11371579285855222445428509042, 3.63218376032489798597574207201, 4.83253515755532556749657600380, 5.34204930324312122153102803384, 5.90451972362547634326313493976, 6.26148239397993775233190421295, 6.90437895775839882981654791774, 7.79831504823702382483865253714, 8.209674498885084252064672125274, 8.685097012833614297337989610175, 9.515885990727354254532371328665, 9.648513446796019982492486605924, 9.878178981329807671425407958060

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