Properties

Label 4-338688-1.1-c1e2-0-8
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 + 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: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.291768351\)
\(L(\frac12)\) \(\approx\) \(1.291768351\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.925045457299704603116489065928, −8.079277740952760928945829672096, −7.79767231908533927131642959225, −7.32162986839021161513702366232, −7.00255343044110607173025669696, −6.37247888227728798454797289956, −5.81505070643734415933143757401, −5.46167295341240608742547978354, −4.84778883354414629110762089589, −4.55847983479225625423580151782, −3.74996468035003867898347641963, −3.16913179166411359156679433175, −2.51551692446386328944873714774, −1.68169268545289806799009243015, −0.68580780236667830437631697436, 0.68580780236667830437631697436, 1.68169268545289806799009243015, 2.51551692446386328944873714774, 3.16913179166411359156679433175, 3.74996468035003867898347641963, 4.55847983479225625423580151782, 4.84778883354414629110762089589, 5.46167295341240608742547978354, 5.81505070643734415933143757401, 6.37247888227728798454797289956, 7.00255343044110607173025669696, 7.32162986839021161513702366232, 7.79767231908533927131642959225, 8.079277740952760928945829672096, 8.925045457299704603116489065928

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