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Properties

Label 4-50e2-1.1-c1e2-0-1
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $0.159402$
Root an. cond. $0.631863$
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  + 4-s − 5·9-s − 6·11-s + 16-s + 10·19-s + 4·31-s − 5·36-s − 6·41-s − 6·44-s − 10·49-s + 4·61-s + 64-s + 24·71-s + 10·76-s − 20·79-s + 16·81-s + 30·89-s + 30·99-s − 36·101-s − 20·109-s + 5·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯
L(s)  = 1  + 1/2·4-s − 5/3·9-s − 1.80·11-s + 1/4·16-s + 2.29·19-s + 0.718·31-s − 5/6·36-s − 0.937·41-s − 0.904·44-s − 1.42·49-s + 0.512·61-s + 1/8·64-s + 2.84·71-s + 1.14·76-s − 2.25·79-s + 16/9·81-s + 3.17·89-s + 3.01·99-s − 3.58·101-s − 1.91·109-s + 5/11·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.159402\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6823642606\)
\(L(\frac12)\) \(\approx\) \(0.6823642606\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.a_z
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−13.31818220322005518851751496924, −12.18462032318833367073561378768, −11.89949732310571442955739052562, −11.16007422368021302024475397710, −10.81431448769284119152476376034, −9.930723730317650982781533521356, −9.447991916742081591691197868932, −8.316812877075225116243141130520, −8.071729218562685355881418106440, −7.29853134793764088505009214313, −6.35306147256574072432179047412, −5.34945783064988058695168525193, −5.20592148788576143134616479765, −3.29699975126484092788987476482, −2.65155040417235216240951299295, 2.65155040417235216240951299295, 3.29699975126484092788987476482, 5.20592148788576143134616479765, 5.34945783064988058695168525193, 6.35306147256574072432179047412, 7.29853134793764088505009214313, 8.071729218562685355881418106440, 8.316812877075225116243141130520, 9.447991916742081591691197868932, 9.930723730317650982781533521356, 10.81431448769284119152476376034, 11.16007422368021302024475397710, 11.89949732310571442955739052562, 12.18462032318833367073561378768, 13.31818220322005518851751496924

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