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Properties

Label 4-760e2-1.1-c1e2-0-11
Degree $4$
Conductor $577600$
Sign $-1$
Analytic cond. $36.8282$
Root an. cond. $2.46345$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·9-s + 12·13-s + 4·17-s + 3·25-s − 4·29-s − 20·37-s − 20·41-s + 4·45-s − 10·49-s + 12·53-s + 4·61-s − 24·65-s − 12·73-s − 5·81-s − 8·85-s + 4·89-s − 36·97-s − 20·101-s + 20·109-s − 20·113-s − 24·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 2/3·9-s + 3.32·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s − 3.28·37-s − 3.12·41-s + 0.596·45-s − 1.42·49-s + 1.64·53-s + 0.512·61-s − 2.97·65-s − 1.40·73-s − 5/9·81-s − 0.867·85-s + 0.423·89-s − 3.65·97-s − 1.99·101-s + 1.91·109-s − 1.88·113-s − 2.21·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(577600\)    =    \(2^{6} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(36.8282\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 577600,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.a_fa
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.a_gg
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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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

−8.294000637754294355036016345474, −8.127953611229209522722729832311, −7.18764194994836978572884806513, −6.78152084585319822916934144821, −6.56710263976437493714298656882, −5.76236805189181119443663467259, −5.46860793773004683410757765520, −5.14121123054718406498749166364, −4.05957201430371047965905503497, −3.86243040340746622434437568454, −3.21566207427278766631032003624, −3.19120987504256316461586726836, −1.68036196072528569368485026712, −1.34946512735630405132580056994, 0, 1.34946512735630405132580056994, 1.68036196072528569368485026712, 3.19120987504256316461586726836, 3.21566207427278766631032003624, 3.86243040340746622434437568454, 4.05957201430371047965905503497, 5.14121123054718406498749166364, 5.46860793773004683410757765520, 5.76236805189181119443663467259, 6.56710263976437493714298656882, 6.78152084585319822916934144821, 7.18764194994836978572884806513, 8.127953611229209522722729832311, 8.294000637754294355036016345474

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