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Properties

Label 4-6e8-1.1-c1e2-0-40
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
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  − 6·11-s − 12·17-s − 8·19-s − 25-s − 6·41-s − 2·43-s − 13·49-s + 6·59-s − 14·67-s − 20·73-s + 18·83-s − 12·89-s + 22·97-s − 24·107-s + 18·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + 173-s + ⋯
L(s)  = 1  − 1.80·11-s − 2.91·17-s − 1.83·19-s − 1/5·25-s − 0.937·41-s − 0.304·43-s − 1.85·49-s + 0.781·59-s − 1.71·67-s − 2.34·73-s + 1.97·83-s − 1.27·89-s + 2.23·97-s − 2.32·107-s + 1.69·113-s + 5/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.92·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.a_bx
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.a_abv
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.67.o_hb
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.a_bl
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.97.aw_md
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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

−7.38675805265098210708587232252, −6.97761942564957656204328929113, −6.41440556897093580869811851316, −6.35198766952899408711193373793, −5.73466891271029423311175361947, −4.96437765868388360434221640221, −4.89580709677700674330960556883, −4.31943704134913367868413741145, −3.95964851266685179088986808677, −3.11209787304952621442346833350, −2.61290832881892085720207796952, −2.15537953527803185525403930602, −1.70581704819924735500239401824, 0, 0, 1.70581704819924735500239401824, 2.15537953527803185525403930602, 2.61290832881892085720207796952, 3.11209787304952621442346833350, 3.95964851266685179088986808677, 4.31943704134913367868413741145, 4.89580709677700674330960556883, 4.96437765868388360434221640221, 5.73466891271029423311175361947, 6.35198766952899408711193373793, 6.41440556897093580869811851316, 6.97761942564957656204328929113, 7.38675805265098210708587232252

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