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Properties

Label 4-560e2-1.1-c1e2-0-40
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
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  + 3-s + 5-s + 5·7-s + 3·9-s + 2·11-s + 15-s − 4·17-s − 2·19-s + 5·21-s + 23-s + 8·27-s + 18·29-s + 4·31-s + 2·33-s + 5·35-s − 4·37-s + 2·41-s − 18·43-s + 3·45-s + 18·49-s − 4·51-s + 10·53-s + 2·55-s − 2·57-s − 10·59-s − 9·61-s + 15·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.88·7-s + 9-s + 0.603·11-s + 0.258·15-s − 0.970·17-s − 0.458·19-s + 1.09·21-s + 0.208·23-s + 1.53·27-s + 3.34·29-s + 0.718·31-s + 0.348·33-s + 0.845·35-s − 0.657·37-s + 0.312·41-s − 2.74·43-s + 0.447·45-s + 18/7·49-s − 0.560·51-s + 1.37·53-s + 0.269·55-s − 0.264·57-s − 1.30·59-s − 1.15·61-s + 1.88·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.560697964\)
\(L(\frac12)\) \(\approx\) \(3.560697964\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_av
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.41.ac_df
43$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.43.s_gl
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.53.ak_bv
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_bp
61$C_2^2$ \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.61.j_u
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.71.bc_na
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_ct
79$C_2^2$ \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_en
83$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.83.w_lb
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
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

−10.75918225059477247521260318799, −10.61400094245281204296903874310, −10.03491452436992801544487347217, −9.901372826978615428649650507682, −8.964172176595218572298651761736, −8.656787668983808565073152161985, −8.438941620644930682805941783654, −8.145345776914433445990037187422, −7.22284430521483368562770377650, −7.15284113747364488551167311343, −6.35473580551789476797455451055, −6.23693928829709708671890587212, −5.08524211226001843257308566396, −4.88178119080158725589911936399, −4.40007195155935534396074380940, −4.07099546771574391429523996449, −2.91669772162565995869674692252, −2.57579544874835584876761979961, −1.44369515060516476287777848657, −1.43813149596822599097363937277, 1.43813149596822599097363937277, 1.44369515060516476287777848657, 2.57579544874835584876761979961, 2.91669772162565995869674692252, 4.07099546771574391429523996449, 4.40007195155935534396074380940, 4.88178119080158725589911936399, 5.08524211226001843257308566396, 6.23693928829709708671890587212, 6.35473580551789476797455451055, 7.15284113747364488551167311343, 7.22284430521483368562770377650, 8.145345776914433445990037187422, 8.438941620644930682805941783654, 8.656787668983808565073152161985, 8.964172176595218572298651761736, 9.901372826978615428649650507682, 10.03491452436992801544487347217, 10.61400094245281204296903874310, 10.75918225059477247521260318799

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