Properties

Degree $4$
Conductor $3111696$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·19-s − 25-s + 3·31-s − 37-s + 2·43-s + 67-s − 3·73-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 3·19-s − 25-s + 3·31-s − 37-s + 2·43-s + 67-s − 3·73-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.343485828\)
\(L(\frac12)\) \(\approx\) \(1.343485828\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.687184356593749704493411338288, −9.275247870096338604079306453823, −9.081666323979300436048649978424, −8.403023092783124412015173049755, −7.972400335505062075757378460648, −7.891062340729175464212660120786, −7.18860862345448173196603323893, −7.11930821621898335104347898863, −6.49898595070475263277650220619, −6.02399397173807095499728119634, −5.53021260668522422814805144838, −5.42507612909214849578848355285, −4.62941671262291896795223466798, −4.49610482149762231237602058076, −3.70144622381404672599981968592, −3.41057191562855977404957052690, −2.61058847721582603051547490613, −2.60865273860038604852418654443, −1.39681928976638470793928278990, −1.04076825266703248020012614521, 1.04076825266703248020012614521, 1.39681928976638470793928278990, 2.60865273860038604852418654443, 2.61058847721582603051547490613, 3.41057191562855977404957052690, 3.70144622381404672599981968592, 4.49610482149762231237602058076, 4.62941671262291896795223466798, 5.42507612909214849578848355285, 5.53021260668522422814805144838, 6.02399397173807095499728119634, 6.49898595070475263277650220619, 7.11930821621898335104347898863, 7.18860862345448173196603323893, 7.891062340729175464212660120786, 7.972400335505062075757378460648, 8.403023092783124412015173049755, 9.081666323979300436048649978424, 9.275247870096338604079306453823, 9.687184356593749704493411338288

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