Properties

Label 4-12e6-1.1-c1e2-0-6
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $190.388$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 6·7-s + 3·11-s + 4·13-s + 19·25-s − 6·29-s − 36·35-s − 4·37-s + 9·41-s + 9·43-s − 12·47-s + 17·49-s − 18·55-s + 15·59-s + 8·61-s − 24·65-s − 15·67-s − 12·71-s − 22·73-s + 18·77-s − 6·79-s − 12·83-s + 24·91-s − 13·97-s + 18·101-s − 24·103-s − 6·107-s + ⋯
L(s)  = 1  − 2.68·5-s + 2.26·7-s + 0.904·11-s + 1.10·13-s + 19/5·25-s − 1.11·29-s − 6.08·35-s − 0.657·37-s + 1.40·41-s + 1.37·43-s − 1.75·47-s + 17/7·49-s − 2.42·55-s + 1.95·59-s + 1.02·61-s − 2.97·65-s − 1.83·67-s − 1.42·71-s − 2.57·73-s + 2.05·77-s − 0.675·79-s − 1.31·83-s + 2.51·91-s − 1.31·97-s + 1.79·101-s − 2.36·103-s − 0.580·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(190.388\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.676489849\)
\(L(\frac12)\) \(\approx\) \(1.676489849\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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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

−9.361323740474410189530340983345, −8.778613092224921971322541929313, −8.578960033804922305298965296241, −8.308283665737431998408192838659, −8.049539434152478492714760760822, −7.50373149729682234945187166908, −7.31490417935710212438231643071, −7.08829664331139679340406510301, −6.39265834029413058026816152693, −5.61869022601222560706319362001, −5.58693448686545831962549647378, −4.71840143464884241617571527404, −4.32362074314574855757476773811, −4.23781989631598703100244333435, −3.83578866984023447702394057208, −3.30426182089901496200839122497, −2.71865113003289787970637671806, −1.64124109278889752664842472454, −1.41598129883655030631448933454, −0.54010337677548023856421021444, 0.54010337677548023856421021444, 1.41598129883655030631448933454, 1.64124109278889752664842472454, 2.71865113003289787970637671806, 3.30426182089901496200839122497, 3.83578866984023447702394057208, 4.23781989631598703100244333435, 4.32362074314574855757476773811, 4.71840143464884241617571527404, 5.58693448686545831962549647378, 5.61869022601222560706319362001, 6.39265834029413058026816152693, 7.08829664331139679340406510301, 7.31490417935710212438231643071, 7.50373149729682234945187166908, 8.049539434152478492714760760822, 8.308283665737431998408192838659, 8.578960033804922305298965296241, 8.778613092224921971322541929313, 9.361323740474410189530340983345

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