Properties

Label 4-246924-1.1-c1e2-0-4
Degree $4$
Conductor $246924$
Sign $-1$
Analytic cond. $15.7440$
Root an. cond. $1.99195$
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  − 2·2-s + 3-s + 3·4-s − 2·6-s − 2·7-s − 4·8-s − 2·9-s + 3·12-s + 4·14-s + 5·16-s + 4·18-s + 19-s − 2·21-s − 4·24-s − 10·25-s − 5·27-s − 6·28-s + 18·29-s − 6·32-s − 6·36-s − 2·38-s + 4·42-s + 16·43-s + 5·48-s − 11·49-s + 20·50-s − 6·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 0.755·7-s − 1.41·8-s − 2/3·9-s + 0.866·12-s + 1.06·14-s + 5/4·16-s + 0.942·18-s + 0.229·19-s − 0.436·21-s − 0.816·24-s − 2·25-s − 0.962·27-s − 1.13·28-s + 3.34·29-s − 1.06·32-s − 36-s − 0.324·38-s + 0.617·42-s + 2.43·43-s + 0.721·48-s − 1.57·49-s + 2.82·50-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 246924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 246924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(246924\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(15.7440\)
Root analytic conductor: \(1.99195\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 246924,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
19$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.684413720393242419277779815238, −8.391177552243354804348940653002, −7.73591843742988285704805588989, −7.72023666635394736173695745964, −6.85841529930894935996514770633, −6.52547725848872771498338713632, −5.81027585652647529822118508915, −5.77798205728261117317953598536, −4.61559374588543556536361618627, −4.07392909124551760489549274271, −3.01249692314631579373849129643, −2.99432139105097413830284496966, −2.19179284747142586381309074581, −1.22349109094330361666806860431, 0, 1.22349109094330361666806860431, 2.19179284747142586381309074581, 2.99432139105097413830284496966, 3.01249692314631579373849129643, 4.07392909124551760489549274271, 4.61559374588543556536361618627, 5.77798205728261117317953598536, 5.81027585652647529822118508915, 6.52547725848872771498338713632, 6.85841529930894935996514770633, 7.72023666635394736173695745964, 7.73591843742988285704805588989, 8.391177552243354804348940653002, 8.684413720393242419277779815238

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