Properties

Label 4-84e2-1.1-c0e2-0-0
Degree $4$
Conductor $7056$
Sign $1$
Analytic cond. $0.00175740$
Root an. cond. $0.204747$
Motivic weight $0$
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 − 7-s − 2·13-s + 19-s + 21-s − 25-s + 27-s + 31-s + 37-s + 2·39-s − 2·43-s − 57-s − 2·61-s + 67-s + 73-s + 75-s + 79-s − 81-s + 2·91-s − 93-s + 4·97-s + 103-s + 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯
L(s)  = 1  − 3-s − 7-s − 2·13-s + 19-s + 21-s − 25-s + 27-s + 31-s + 37-s + 2·39-s − 2·43-s − 57-s − 2·61-s + 67-s + 73-s + 75-s + 79-s − 81-s + 2·91-s − 93-s + 4·97-s + 103-s + 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.00175740\)
Root analytic conductor: \(0.204747\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{84} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7056,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2070897827\)
\(L(\frac12)\) \(\approx\) \(0.2070897827\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_1$ \( ( 1 - T )^{4} \)
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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

−14.74600246740631530719188745074, −14.33197631703146060715751911730, −13.63570593099300485737171669080, −13.20498120399190008275121892141, −12.33651690769049716182029767608, −12.27926998190461195602564601842, −11.57081641310592920438165715557, −11.28616606617349332842095808165, −10.11348932277312344731628512468, −10.10977840826758241325303230439, −9.514310179830675798001626081592, −8.796010299104171695587299127572, −7.74268924650426005669373898019, −7.44725460239841168931737259170, −6.33268371238830377081639818438, −6.30895696473354994681334896900, −5.07920460323713432335456360487, −4.89098280278214127535463544377, −3.52084052442906496218604031148, −2.56497434085969418732679777437, 2.56497434085969418732679777437, 3.52084052442906496218604031148, 4.89098280278214127535463544377, 5.07920460323713432335456360487, 6.30895696473354994681334896900, 6.33268371238830377081639818438, 7.44725460239841168931737259170, 7.74268924650426005669373898019, 8.796010299104171695587299127572, 9.514310179830675798001626081592, 10.10977840826758241325303230439, 10.11348932277312344731628512468, 11.28616606617349332842095808165, 11.57081641310592920438165715557, 12.27926998190461195602564601842, 12.33651690769049716182029767608, 13.20498120399190008275121892141, 13.63570593099300485737171669080, 14.33197631703146060715751911730, 14.74600246740631530719188745074

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