Properties

Label 4-57e2-1.1-c0e2-0-0
Degree $4$
Conductor $3249$
Sign $1$
Analytic cond. $0.000809215$
Root an. cond. $0.168661$
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 − 4-s − 2·7-s + 12-s + 13-s + 2·19-s + 2·21-s − 25-s + 27-s + 2·28-s − 2·31-s − 2·37-s − 39-s + 43-s + 49-s − 52-s − 2·57-s + 61-s + 64-s + 67-s + 73-s + 75-s − 2·76-s + 79-s − 81-s − 2·84-s − 2·91-s + ⋯
L(s)  = 1  − 3-s − 4-s − 2·7-s + 12-s + 13-s + 2·19-s + 2·21-s − 25-s + 27-s + 2·28-s − 2·31-s − 2·37-s − 39-s + 43-s + 49-s − 52-s − 2·57-s + 61-s + 64-s + 67-s + 73-s + 75-s − 2·76-s + 79-s − 81-s − 2·84-s − 2·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.000809215\)
Root analytic conductor: \(0.168661\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{57} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3249,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1395290986\)
\(L(\frac12)\) \(\approx\) \(0.1395290986\)
\(L(1)\) 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)$
bad3$C_2$ \( 1 + T + T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−15.78158863451418310600567398366, −15.73698933792598545940443005947, −14.50422369454586197462514135495, −13.91118790105346893717398966072, −13.52036331503366893986103235457, −13.04786003793567616204031954427, −12.40570419368847873598634726769, −12.01208341174308721549737477039, −11.16295984834053954222603330510, −10.73867704030288327198980522926, −9.755934096435888961389015983965, −9.530714621167639954748207724266, −8.979906840158385872094533206307, −8.113025414063899120618495471630, −7.02046440325267567096761130699, −6.56213261943062567216093815027, −5.50846861495591810506841875830, −5.42600079494021974315876790764, −3.89339273183383705579882973780, −3.30843370639598070011740501216, 3.30843370639598070011740501216, 3.89339273183383705579882973780, 5.42600079494021974315876790764, 5.50846861495591810506841875830, 6.56213261943062567216093815027, 7.02046440325267567096761130699, 8.113025414063899120618495471630, 8.979906840158385872094533206307, 9.530714621167639954748207724266, 9.755934096435888961389015983965, 10.73867704030288327198980522926, 11.16295984834053954222603330510, 12.01208341174308721549737477039, 12.40570419368847873598634726769, 13.04786003793567616204031954427, 13.52036331503366893986103235457, 13.91118790105346893717398966072, 14.50422369454586197462514135495, 15.73698933792598545940443005947, 15.78158863451418310600567398366

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