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Properties

Label 4-338688-1.1-c1e2-0-75
Degree $4$
Conductor $338688$
Sign $-1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·13-s − 8·19-s − 2·25-s + 27-s + 4·37-s − 4·39-s + 49-s − 8·57-s − 4·61-s − 12·73-s − 2·75-s − 16·79-s + 81-s + 20·97-s − 4·109-s + 4·111-s − 4·117-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.83·19-s − 2/5·25-s + 0.192·27-s + 0.657·37-s − 0.640·39-s + 1/7·49-s − 1.05·57-s − 0.512·61-s − 1.40·73-s − 0.230·75-s − 1.80·79-s + 1/9·81-s + 2.03·97-s − 0.383·109-s + 0.379·111-s − 0.369·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0824·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.e_o
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.i_bm
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.41.a_aw
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.53.a_acg
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.e_as
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.71.a_co
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.83.a_aec
89$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.89.a_ec
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−8.545196259939831190604891203096, −8.084817426260425380470468261288, −7.57828293431928156633454079856, −7.27712339215461453439044534379, −6.66076576256304296160190655054, −6.21666319901158747800445457771, −5.73813628904751972112835174526, −5.00868379143768967661814563221, −4.52361602173296863802803138278, −4.12019922721860777717659742162, −3.47555149106795526584770899468, −2.64215730531238732386170290577, −2.32677321564199613013048671795, −1.47571293091629992053184662728, 0, 1.47571293091629992053184662728, 2.32677321564199613013048671795, 2.64215730531238732386170290577, 3.47555149106795526584770899468, 4.12019922721860777717659742162, 4.52361602173296863802803138278, 5.00868379143768967661814563221, 5.73813628904751972112835174526, 6.21666319901158747800445457771, 6.66076576256304296160190655054, 7.27712339215461453439044534379, 7.57828293431928156633454079856, 8.084817426260425380470468261288, 8.545196259939831190604891203096

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