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Properties

Label 4-2496e2-1.1-c1e2-0-22
Degree $4$
Conductor $6230016$
Sign $1$
Analytic cond. $397.231$
Root an. cond. $4.46437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 6·13-s − 4·17-s − 16·23-s + 6·25-s − 4·27-s − 4·29-s + 12·39-s + 16·43-s − 2·49-s + 8·51-s − 12·53-s − 4·61-s + 32·69-s − 12·75-s + 5·81-s + 8·87-s + 20·101-s − 8·103-s − 40·107-s − 20·113-s − 18·117-s − 14·121-s + 127-s − 32·129-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 1.66·13-s − 0.970·17-s − 3.33·23-s + 6/5·25-s − 0.769·27-s − 0.742·29-s + 1.92·39-s + 2.43·43-s − 2/7·49-s + 1.12·51-s − 1.64·53-s − 0.512·61-s + 3.85·69-s − 1.38·75-s + 5/9·81-s + 0.857·87-s + 1.99·101-s − 0.788·103-s − 3.86·107-s − 1.88·113-s − 1.66·117-s − 1.27·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6230016\)    =    \(2^{12} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(397.231\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6230016,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.41.a_ada
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.59.a_aek
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.83.a_be
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−8.774122938092192689635398306741, −8.198170640875879483694706895053, −7.80027620533378917643787793897, −7.68922444124778802244852055201, −7.11211347056595625423405606823, −6.78042750591045848011188619415, −6.32774576011944122046733324478, −6.08765278721864012664090033179, −5.47370695613666198886768978156, −5.41303236614920804770108968450, −4.66241738282695601451822942507, −4.40169823233251598748663983333, −4.15325793583821803609865162559, −3.55168880619980416170380306026, −2.78600508295564923287730164033, −2.28055563812124620616158401142, −1.94421490093683546786349999249, −1.15430446050298684736808559259, 0, 0, 1.15430446050298684736808559259, 1.94421490093683546786349999249, 2.28055563812124620616158401142, 2.78600508295564923287730164033, 3.55168880619980416170380306026, 4.15325793583821803609865162559, 4.40169823233251598748663983333, 4.66241738282695601451822942507, 5.41303236614920804770108968450, 5.47370695613666198886768978156, 6.08765278721864012664090033179, 6.32774576011944122046733324478, 6.78042750591045848011188619415, 7.11211347056595625423405606823, 7.68922444124778802244852055201, 7.80027620533378917643787793897, 8.198170640875879483694706895053, 8.774122938092192689635398306741

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