Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·13-s − 19-s + 5·25-s + 11·31-s − 11·37-s − 26·43-s + 14·61-s − 5·67-s + 17·73-s − 17·79-s − 28·97-s − 13·103-s + 19·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.77·13-s − 0.229·19-s + 25-s + 1.97·31-s − 1.80·37-s − 3.96·43-s + 1.79·61-s − 0.610·67-s + 1.98·73-s − 1.91·79-s − 2.84·97-s − 1.28·103-s + 1.81·109-s + 121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.8324675234\)
\(L(\frac12)\) \(\approx\) \(0.8324675234\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.733484805010312995757072075307, −9.065661880740095282906361013185, −8.647372999198639884868872893249, −8.255530357163612598956641288302, −8.080852764483335655557793315926, −7.44085467768578459224412039613, −6.99228793999434855221368045232, −6.75045881725246026399194981917, −6.61471505043786677071393997187, −5.79784314409085780556131012788, −5.19829198354000330994047648204, −4.89580628301305627995229613751, −4.88594222999733455436622366586, −4.14643365204930599716803810902, −3.56340001883532626000430154013, −2.90917606892586871300613969247, −2.68442837366993884113119511113, −2.03418938979461441867474811686, −1.43413597535384233558640062636, −0.33433785379799074607024525153, 0.33433785379799074607024525153, 1.43413597535384233558640062636, 2.03418938979461441867474811686, 2.68442837366993884113119511113, 2.90917606892586871300613969247, 3.56340001883532626000430154013, 4.14643365204930599716803810902, 4.88594222999733455436622366586, 4.89580628301305627995229613751, 5.19829198354000330994047648204, 5.79784314409085780556131012788, 6.61471505043786677071393997187, 6.75045881725246026399194981917, 6.99228793999434855221368045232, 7.44085467768578459224412039613, 8.080852764483335655557793315926, 8.255530357163612598956641288302, 8.647372999198639884868872893249, 9.065661880740095282906361013185, 9.733484805010312995757072075307

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