Properties

Degree $4$
Conductor $9144576$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 4·7-s + 12·17-s + 17·25-s + 24·35-s + 2·37-s + 6·41-s − 20·43-s − 12·47-s + 9·49-s + 12·59-s − 4·67-s − 28·79-s + 12·83-s − 72·85-s − 18·89-s − 36·101-s − 22·109-s − 48·119-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2.68·5-s − 1.51·7-s + 2.91·17-s + 17/5·25-s + 4.05·35-s + 0.328·37-s + 0.937·41-s − 3.04·43-s − 1.75·47-s + 9/7·49-s + 1.56·59-s − 0.488·67-s − 3.15·79-s + 1.31·83-s − 7.80·85-s − 1.90·89-s − 3.58·101-s − 2.10·109-s − 4.40·119-s − 0.454·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07944426350\)
\(L(\frac12)\) \(\approx\) \(0.07944426350\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−8.782123655175865638387133269304, −8.291912165078873982046773746583, −8.091557451854792137476252105068, −7.81643054515517726610614443296, −7.60277070486980131044180321839, −6.99219211832104919156461737014, −6.75475957125653173449070368729, −6.54681280671259737121490475387, −5.72818608844261016504183430292, −5.48290297729960953092363597728, −5.12754387913780638659933960935, −4.40914325042995461081554599157, −4.02357223526393147613956281645, −3.77688240111925188310556803731, −3.35688742724583589401919427414, −2.94561479475134306526077932657, −2.88869269192788205398542592838, −1.54825719913142585626980472826, −1.01932724016355281992637817179, −0.10897078076687579224833851951, 0.10897078076687579224833851951, 1.01932724016355281992637817179, 1.54825719913142585626980472826, 2.88869269192788205398542592838, 2.94561479475134306526077932657, 3.35688742724583589401919427414, 3.77688240111925188310556803731, 4.02357223526393147613956281645, 4.40914325042995461081554599157, 5.12754387913780638659933960935, 5.48290297729960953092363597728, 5.72818608844261016504183430292, 6.54681280671259737121490475387, 6.75475957125653173449070368729, 6.99219211832104919156461737014, 7.60277070486980131044180321839, 7.81643054515517726610614443296, 8.091557451854792137476252105068, 8.291912165078873982046773746583, 8.782123655175865638387133269304

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