Properties

Label 4-60e3-1.1-c1e2-0-26
Degree $4$
Conductor $216000$
Sign $-1$
Analytic cond. $13.7723$
Root an. cond. $1.92642$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 2·9-s − 2·10-s − 2·12-s + 3·13-s + 15-s − 4·16-s − 4·18-s − 2·20-s + 25-s + 6·26-s + 5·27-s + 2·30-s − 2·31-s − 8·32-s − 4·36-s − 12·37-s − 3·39-s − 3·41-s + 3·43-s + 2·45-s + 4·48-s − 10·49-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 2/3·9-s − 0.632·10-s − 0.577·12-s + 0.832·13-s + 0.258·15-s − 16-s − 0.942·18-s − 0.447·20-s + 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.365·30-s − 0.359·31-s − 1.41·32-s − 2/3·36-s − 1.97·37-s − 0.480·39-s − 0.468·41-s + 0.457·43-s + 0.298·45-s + 0.577·48-s − 1.42·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(13.7723\)
Root analytic conductor: \(1.92642\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 216000,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - p T + p T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
47$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 176 T^{2} + p^{2} T^{4} \)
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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

−8.722339061279030502026217557080, −8.326625773390273821916688279470, −7.86104944075813803200111539928, −6.98306956144003059227349427373, −6.80029561603651451983469425798, −6.20925779024294472317126229619, −5.74294810853403922092955208519, −5.32290141168043296240714248598, −4.84739726754712687571159778127, −4.31296215270085071722819888265, −3.57959421488730987783136762088, −3.31251437718625774347934014906, −2.57128120301065922656290658012, −1.55179478107478001093121490033, 0, 1.55179478107478001093121490033, 2.57128120301065922656290658012, 3.31251437718625774347934014906, 3.57959421488730987783136762088, 4.31296215270085071722819888265, 4.84739726754712687571159778127, 5.32290141168043296240714248598, 5.74294810853403922092955208519, 6.20925779024294472317126229619, 6.80029561603651451983469425798, 6.98306956144003059227349427373, 7.86104944075813803200111539928, 8.326625773390273821916688279470, 8.722339061279030502026217557080

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