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Properties

Label 4-338688-1.1-c1e2-0-43
Degree $4$
Conductor $338688$
Sign $1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
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 + 8·5-s + 9-s − 8·15-s − 8·19-s + 4·23-s + 38·25-s − 27-s − 4·29-s − 8·43-s + 8·45-s + 24·47-s + 49-s − 12·53-s + 8·57-s − 16·67-s − 4·69-s + 28·71-s − 4·73-s − 38·75-s + 81-s + 4·87-s − 64·95-s − 4·97-s + 32·101-s + 32·115-s − 18·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 3.57·5-s + 1/3·9-s − 2.06·15-s − 1.83·19-s + 0.834·23-s + 38/5·25-s − 0.192·27-s − 0.742·29-s − 1.21·43-s + 1.19·45-s + 3.50·47-s + 1/7·49-s − 1.64·53-s + 1.05·57-s − 1.95·67-s − 0.481·69-s + 3.32·71-s − 0.468·73-s − 4.38·75-s + 1/9·81-s + 0.428·87-s − 6.56·95-s − 0.406·97-s + 3.18·101-s + 2.98·115-s − 1.63·121-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.088585436\)
\(L(\frac12)\) \(\approx\) \(3.088585436\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.5.ai_ba
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.71.abc_na
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−9.069101250737246906911171178090, −8.621234568783701478950617886590, −7.79195451575718587485184031739, −6.96207355783957131001717701468, −6.78902576466258518849814546939, −6.06564976659535992293108524760, −6.05700480174216171276936327443, −5.62370507273544228914734043068, −4.91059247592313575518339819369, −4.80916909916633799382690392302, −3.78443484852293199292674683103, −2.79378519992244683258213909080, −2.24239796255929817450421188994, −1.88896973398279373978653204444, −1.12644399783657185408249656998, 1.12644399783657185408249656998, 1.88896973398279373978653204444, 2.24239796255929817450421188994, 2.79378519992244683258213909080, 3.78443484852293199292674683103, 4.80916909916633799382690392302, 4.91059247592313575518339819369, 5.62370507273544228914734043068, 6.05700480174216171276936327443, 6.06564976659535992293108524760, 6.78902576466258518849814546939, 6.96207355783957131001717701468, 7.79195451575718587485184031739, 8.621234568783701478950617886590, 9.069101250737246906911171178090

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