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Properties

Label 4-4080e2-1.1-c1e2-0-11
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
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  − 9-s − 12·13-s − 8·17-s + 8·19-s − 25-s + 12·43-s + 16·47-s + 14·49-s + 24·59-s − 4·67-s + 81-s + 4·89-s + 4·101-s + 12·103-s + 12·117-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + 163-s + 167-s + 82·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 3.32·13-s − 1.94·17-s + 1.83·19-s − 1/5·25-s + 1.82·43-s + 2.33·47-s + 2·49-s + 3.12·59-s − 0.488·67-s + 1/9·81-s + 0.423·89-s + 0.398·101-s + 1.18·103-s + 1.10·117-s + 1.63·121-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 + 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.788532738\)
\(L(\frac12)\) \(\approx\) \(1.788532738\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 8 T + p T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.41.a_ada
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \) 2.71.a_afi
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.79.a_du
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.97.a_c
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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

−8.537268689021860775209082545411, −8.458600485785885237736380394264, −7.48551592212459773966161985045, −7.44262541810252015290754727739, −7.23290837446301258104756426354, −7.17271696475398337302626623281, −6.38882164229318664348890395122, −6.03347387545744731407350129143, −5.45655678795182600124588498939, −5.33949716550573148862964394464, −4.85229505788858902625068042586, −4.58447673569656653760935066916, −3.96931179942362846023008738053, −3.86074091016959400225736932836, −2.84743911237637561084788792504, −2.69948141339073715684692239417, −2.23169803119502957961992110467, −2.08703455131434809973831977406, −0.853354919044897850875641570264, −0.48310783883612080792054410485, 0.48310783883612080792054410485, 0.853354919044897850875641570264, 2.08703455131434809973831977406, 2.23169803119502957961992110467, 2.69948141339073715684692239417, 2.84743911237637561084788792504, 3.86074091016959400225736932836, 3.96931179942362846023008738053, 4.58447673569656653760935066916, 4.85229505788858902625068042586, 5.33949716550573148862964394464, 5.45655678795182600124588498939, 6.03347387545744731407350129143, 6.38882164229318664348890395122, 7.17271696475398337302626623281, 7.23290837446301258104756426354, 7.44262541810252015290754727739, 7.48551592212459773966161985045, 8.458600485785885237736380394264, 8.537268689021860775209082545411

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