Properties

Label 4-57e2-1.1-c1e2-0-1
Degree $4$
Conductor $3249$
Sign $-1$
Analytic cond. $0.207159$
Root an. cond. $0.674646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·4-s − 2·7-s + 9-s + 8·12-s − 8·13-s + 12·16-s + 2·19-s + 4·21-s − 25-s + 4·27-s + 8·28-s − 8·31-s − 4·36-s + 4·37-s + 16·39-s − 2·43-s − 24·48-s − 11·49-s + 32·52-s − 4·57-s − 2·61-s − 2·63-s − 32·64-s − 8·67-s − 14·73-s + 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 2·4-s − 0.755·7-s + 1/3·9-s + 2.30·12-s − 2.21·13-s + 3·16-s + 0.458·19-s + 0.872·21-s − 1/5·25-s + 0.769·27-s + 1.51·28-s − 1.43·31-s − 2/3·36-s + 0.657·37-s + 2.56·39-s − 0.304·43-s − 3.46·48-s − 1.57·49-s + 4.43·52-s − 0.529·57-s − 0.256·61-s − 0.251·63-s − 4·64-s − 0.977·67-s − 1.63·73-s + 0.230·75-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(0.207159\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3249} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3249,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−12.45987383937424874038080001140, −12.06296001644352266090310246809, −11.37237044309941020825023181952, −10.24320445923116880350231460909, −10.15333178148614777799832547132, −9.238767264318381906771348588364, −9.180168549070062598241568188033, −7.968751144610877316457317025190, −7.43141403312000422638829977686, −6.39084306102520193435942181609, −5.51097905283736056328679977781, −5.03912355415234199771433602492, −4.41466206651142691425988858089, −3.23857552173567549926694025031, 0, 3.23857552173567549926694025031, 4.41466206651142691425988858089, 5.03912355415234199771433602492, 5.51097905283736056328679977781, 6.39084306102520193435942181609, 7.43141403312000422638829977686, 7.968751144610877316457317025190, 9.180168549070062598241568188033, 9.238767264318381906771348588364, 10.15333178148614777799832547132, 10.24320445923116880350231460909, 11.37237044309941020825023181952, 12.06296001644352266090310246809, 12.45987383937424874038080001140

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