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Properties

Label 4-2200e2-1.1-c1e2-0-8
Degree $4$
Conductor $4840000$
Sign $1$
Analytic cond. $308.602$
Root an. cond. $4.19131$
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  + 6·9-s − 2·11-s − 8·19-s + 4·29-s + 16·31-s + 20·41-s − 2·49-s + 8·59-s − 4·61-s + 32·79-s + 27·81-s + 12·89-s − 12·99-s − 20·101-s + 4·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 48·171-s + ⋯
L(s)  = 1  + 2·9-s − 0.603·11-s − 1.83·19-s + 0.742·29-s + 2.87·31-s + 3.12·41-s − 2/7·49-s + 1.04·59-s − 0.512·61-s + 3.60·79-s + 3·81-s + 1.27·89-s − 1.20·99-s − 1.99·101-s + 0.383·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s − 3.67·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4840000\)    =    \(2^{6} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(308.602\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.209945727\)
\(L(\frac12)\) \(\approx\) \(3.209945727\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
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

−9.173272638906922952524654856104, −9.045924795057530324134240981637, −8.201112689554953730397912253755, −8.187080981213756538563395227620, −7.60170240171437640692729771104, −7.56490432803600403795294767739, −6.64546130247805982764041713866, −6.58147224282461997045853947005, −6.38576810848960751697828259561, −5.78284919084614804057441066895, −5.08670282851686742058383987165, −4.78788613270423491250993019853, −4.24930679036380430135448509784, −4.22966920562439457284578760930, −3.64322421927850082378939775402, −2.81849396306739841050034806558, −2.41471767067084116433035817825, −2.03003099452872654042095021579, −1.13433667192347341342230980392, −0.73052228231033222656394211521, 0.73052228231033222656394211521, 1.13433667192347341342230980392, 2.03003099452872654042095021579, 2.41471767067084116433035817825, 2.81849396306739841050034806558, 3.64322421927850082378939775402, 4.22966920562439457284578760930, 4.24930679036380430135448509784, 4.78788613270423491250993019853, 5.08670282851686742058383987165, 5.78284919084614804057441066895, 6.38576810848960751697828259561, 6.58147224282461997045853947005, 6.64546130247805982764041713866, 7.56490432803600403795294767739, 7.60170240171437640692729771104, 8.187080981213756538563395227620, 8.201112689554953730397912253755, 9.045924795057530324134240981637, 9.173272638906922952524654856104

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