Properties

Label 4-338688-1.1-c1e2-0-14
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 + 9-s − 4·13-s − 6·25-s + 27-s + 8·31-s − 4·37-s − 4·39-s + 8·43-s + 49-s + 4·61-s + 12·73-s − 6·75-s + 24·79-s + 81-s + 8·93-s + 12·97-s + 24·103-s − 4·109-s − 4·111-s − 4·117-s + 18·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.10·13-s − 6/5·25-s + 0.192·27-s + 1.43·31-s − 0.657·37-s − 0.640·39-s + 1.21·43-s + 1/7·49-s + 0.512·61-s + 1.40·73-s − 0.692·75-s + 2.70·79-s + 1/9·81-s + 0.829·93-s + 1.21·97-s + 2.36·103-s − 0.383·109-s − 0.379·111-s − 0.369·117-s + 1.63·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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

\(L(1)\) \(\approx\) \(2.003284843\)
\(L(\frac12)\) \(\approx\) \(2.003284843\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−8.706564464137751069200249298750, −8.240602488938245802016396217227, −7.87376874875125307911625056574, −7.40711330576084882407802401038, −7.03863844081642469018514999252, −6.39181082441455792332955324303, −6.00739949098589577520102438401, −5.32183972856988414502913581468, −4.81025107036276944964430444960, −4.37155468203008880327409875901, −3.67942876990573876562030827181, −3.19874738072533047732671001813, −2.32855807251337373014471552520, −2.06800903441179660350792541068, −0.77837018349240527020268404305, 0.77837018349240527020268404305, 2.06800903441179660350792541068, 2.32855807251337373014471552520, 3.19874738072533047732671001813, 3.67942876990573876562030827181, 4.37155468203008880327409875901, 4.81025107036276944964430444960, 5.32183972856988414502913581468, 6.00739949098589577520102438401, 6.39181082441455792332955324303, 7.03863844081642469018514999252, 7.40711330576084882407802401038, 7.87376874875125307911625056574, 8.240602488938245802016396217227, 8.706564464137751069200249298750

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