Properties

Label 4-528e2-1.1-c1e2-0-6
Degree $4$
Conductor $278784$
Sign $1$
Analytic cond. $17.7755$
Root an. cond. $2.05331$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s − 25-s + 5·27-s − 10·31-s − 14·37-s + 14·49-s + 26·67-s + 75-s + 81-s + 10·93-s + 34·97-s + 8·103-s + 14·111-s − 11·121-s + 127-s + 131-s + 137-s + 139-s − 14·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 1/5·25-s + 0.962·27-s − 1.79·31-s − 2.30·37-s + 2·49-s + 3.17·67-s + 0.115·75-s + 1/9·81-s + 1.03·93-s + 3.45·97-s + 0.788·103-s + 1.32·111-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.15·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(17.7755\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.003699148\)
\(L(\frac12)\) \(\approx\) \(1.003699148\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−10.97935125414072199391846000303, −10.61967844044547947219519856922, −10.43604857520895397313552234020, −9.761033227496527930096902990677, −9.268737331075924840089156276012, −8.792118286725997175229462966309, −8.581527218580175625569397603300, −7.898670745427555371624584052664, −7.46155897196190805612781753676, −6.79421309109509854098817947383, −6.69743043306837242351934064057, −5.71868958286481369615025403707, −5.64440029760153032627169184873, −5.10269933282505063566711883265, −4.53277226051154901591940858948, −3.58599230319996879614848463687, −3.51364232222996827586422714352, −2.43257441357989869742041920563, −1.85953456650108989863298346040, −0.60389014896254339106856344580, 0.60389014896254339106856344580, 1.85953456650108989863298346040, 2.43257441357989869742041920563, 3.51364232222996827586422714352, 3.58599230319996879614848463687, 4.53277226051154901591940858948, 5.10269933282505063566711883265, 5.64440029760153032627169184873, 5.71868958286481369615025403707, 6.69743043306837242351934064057, 6.79421309109509854098817947383, 7.46155897196190805612781753676, 7.898670745427555371624584052664, 8.581527218580175625569397603300, 8.792118286725997175229462966309, 9.268737331075924840089156276012, 9.761033227496527930096902990677, 10.43604857520895397313552234020, 10.61967844044547947219519856922, 10.97935125414072199391846000303

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