Properties

Label 4-627e2-1.1-c1e2-0-12
Degree $4$
Conductor $393129$
Sign $-1$
Analytic cond. $25.0662$
Root an. cond. $2.23754$
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  + 4·2-s + 3-s + 8·4-s + 4·6-s − 4·7-s + 8·8-s − 2·9-s + 8·12-s − 16·14-s − 4·16-s − 8·18-s − 4·21-s + 8·24-s − 9·25-s − 5·27-s − 32·28-s − 32·32-s − 16·36-s + 16·41-s − 16·42-s − 12·43-s − 4·48-s − 2·49-s − 36·50-s + 12·53-s − 20·54-s − 32·56-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.577·3-s + 4·4-s + 1.63·6-s − 1.51·7-s + 2.82·8-s − 2/3·9-s + 2.30·12-s − 4.27·14-s − 16-s − 1.88·18-s − 0.872·21-s + 1.63·24-s − 9/5·25-s − 0.962·27-s − 6.04·28-s − 5.65·32-s − 8/3·36-s + 2.49·41-s − 2.46·42-s − 1.82·43-s − 0.577·48-s − 2/7·49-s − 5.09·50-s + 1.64·53-s − 2.72·54-s − 4.27·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(393129\)    =    \(3^{2} \cdot 11^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(25.0662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 393129,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.228854358433494441874279371750, −7.991382198601661869330644512838, −7.08610298939652805070973785571, −6.66503987304844816686807720890, −6.44705964567155032403967809781, −5.66407660024727617127120629681, −5.65921461221479271613429705311, −5.19096686691774806327844897047, −4.19085115174357929506261869806, −4.09468272193076097429923581775, −3.55069629742260821889174578377, −3.09000916592887094247460325417, −2.62758010025505468758993615534, −2.12638900923174362608185387010, 0, 2.12638900923174362608185387010, 2.62758010025505468758993615534, 3.09000916592887094247460325417, 3.55069629742260821889174578377, 4.09468272193076097429923581775, 4.19085115174357929506261869806, 5.19096686691774806327844897047, 5.65921461221479271613429705311, 5.66407660024727617127120629681, 6.44705964567155032403967809781, 6.66503987304844816686807720890, 7.08610298939652805070973785571, 7.991382198601661869330644512838, 8.228854358433494441874279371750

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