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Properties

Label 4-50000-1.1-c1e2-0-0
Degree $4$
Conductor $50000$
Sign $-1$
Analytic cond. $3.18804$
Root an. cond. $1.33622$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·11-s − 31-s − 11·41-s − 61-s − 31·71-s − 9·81-s + 101-s + 103-s + 107-s + 109-s + 113-s + 70·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 3.31·11-s − 0.179·31-s − 1.71·41-s − 0.128·61-s − 3.67·71-s − 81-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 6.36·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50000\)    =    \(2^{4} \cdot 5^{5}\)
Sign: $-1$
Analytic conductor: \(3.18804\)
Root analytic conductor: \(1.33622\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 50000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_4$ \( 1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.11.l_bz
13$C_2^2$ \( 1 + p^{2} T^{4} \) 2.13.a_a
17$C_2^2$ \( 1 + p^{2} T^{4} \) 2.17.a_a
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + p^{2} T^{4} \) 2.23.a_a
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_4$ \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_abn
37$C_2^2$ \( 1 + p^{2} T^{4} \) 2.37.a_a
41$C_4$ \( 1 + 11 T + 111 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.41.l_eh
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2^2$ \( 1 + p^{2} T^{4} \) 2.53.a_a
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_4$ \( 1 + T - 29 T^{2} + p T^{3} + p^{2} T^{4} \) 2.61.b_abd
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_4$ \( 1 + 31 T + 381 T^{2} + 31 p T^{3} + p^{2} T^{4} \) 2.71.bf_or
73$C_2^2$ \( 1 + p^{2} T^{4} \) 2.73.a_a
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 + p^{2} T^{4} \) 2.97.a_a
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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

−15.0550911686, −14.6460749471, −13.8986178895, −13.4929597632, −13.1806709956, −12.8391000730, −12.3763982805, −11.7109472704, −11.3035748003, −10.5823938099, −10.3657004063, −10.1177472465, −9.39742927222, −8.58742173695, −8.37205110350, −7.64986841047, −7.46763675687, −6.77258940349, −5.83874891319, −5.52578619650, −4.92795946160, −4.42123532946, −3.21951826798, −2.81908159172, −1.94559693000, 0, 1.94559693000, 2.81908159172, 3.21951826798, 4.42123532946, 4.92795946160, 5.52578619650, 5.83874891319, 6.77258940349, 7.46763675687, 7.64986841047, 8.37205110350, 8.58742173695, 9.39742927222, 10.1177472465, 10.3657004063, 10.5823938099, 11.3035748003, 11.7109472704, 12.3763982805, 12.8391000730, 13.1806709956, 13.4929597632, 13.8986178895, 14.6460749471, 15.0550911686

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