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Properties

Label 4-338688-1.1-c1e2-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s + 4·19-s − 2·21-s − 2·25-s − 27-s − 6·29-s − 4·31-s − 4·37-s + 12·47-s − 3·49-s + 18·53-s − 4·57-s + 12·59-s + 2·63-s + 2·75-s + 81-s + 6·87-s + 4·93-s − 4·103-s − 8·109-s + 4·111-s + 18·113-s − 14·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.917·19-s − 0.436·21-s − 2/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.657·37-s + 1.75·47-s − 3/7·49-s + 2.47·53-s − 0.529·57-s + 1.56·59-s + 0.251·63-s + 0.230·75-s + 1/9·81-s + 0.643·87-s + 0.414·93-s − 0.394·103-s − 0.766·109-s + 0.379·111-s + 1.69·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-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: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.554079449\)
\(L(\frac12)\) \(\approx\) \(1.554079449\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.e_be
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.41.a_aw
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.am_dq
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.as_gw
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.am_eo
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.67.a_w
71$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.71.a_acg
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.a_de
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.97.a_ao
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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

−8.736272632803903616725633192144, −8.332754815606835761720677805692, −7.71152294031967869430758183216, −7.24403310731652543940071539382, −7.12253836713531318680101133231, −6.37177674532174965327631059228, −5.69899050063550625067626704490, −5.42303130011221349690163040251, −5.11547796071614781021147659422, −4.22719552159999341516393785478, −3.95306624174800925731894929604, −3.24409401169593907858360144011, −2.33153868590346268044462857856, −1.73210144303801819716232502409, −0.76185106146870641907496167592, 0.76185106146870641907496167592, 1.73210144303801819716232502409, 2.33153868590346268044462857856, 3.24409401169593907858360144011, 3.95306624174800925731894929604, 4.22719552159999341516393785478, 5.11547796071614781021147659422, 5.42303130011221349690163040251, 5.69899050063550625067626704490, 6.37177674532174965327631059228, 7.12253836713531318680101133231, 7.24403310731652543940071539382, 7.71152294031967869430758183216, 8.332754815606835761720677805692, 8.736272632803903616725633192144

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