Properties

Label 4-5760-1.1-c1e2-0-0
Degree $4$
Conductor $5760$
Sign $1$
Analytic cond. $0.367262$
Root an. cond. $0.778474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s − 2·9-s − 2·10-s + 12-s + 2·15-s + 16-s + 2·18-s − 5·19-s + 2·20-s − 3·23-s − 24-s + 2·25-s − 5·27-s + 3·29-s − 2·30-s − 32-s − 2·36-s + 5·38-s − 2·40-s + 7·43-s − 4·45-s + 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.471·18-s − 1.14·19-s + 0.447·20-s − 0.625·23-s − 0.204·24-s + 2/5·25-s − 0.962·27-s + 0.557·29-s − 0.365·30-s − 0.176·32-s − 1/3·36-s + 0.811·38-s − 0.316·40-s + 1.06·43-s − 0.596·45-s + 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5760\)    =    \(2^{7} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.367262\)
Root analytic conductor: \(0.778474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5760,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8084540150\)
\(L(\frac12)\) \(\approx\) \(0.8084540150\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
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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

−11.94879663068125396855491505755, −11.48025403259353863848869431157, −10.76993179663608470720790519061, −10.23892729710416322192518254732, −9.729049277107821360014115576610, −9.121933382765696920995619073396, −8.532832237284381714477548414914, −8.178114840955523972607042414056, −7.33404004829892313575769766146, −6.49078734165607754173981784218, −6.00077677817699022382696369696, −5.18857919569295457660785613666, −4.01289098763868268766009594022, −2.84628359590963579318435033144, −2.00306439532167969849245728601, 2.00306439532167969849245728601, 2.84628359590963579318435033144, 4.01289098763868268766009594022, 5.18857919569295457660785613666, 6.00077677817699022382696369696, 6.49078734165607754173981784218, 7.33404004829892313575769766146, 8.178114840955523972607042414056, 8.532832237284381714477548414914, 9.121933382765696920995619073396, 9.729049277107821360014115576610, 10.23892729710416322192518254732, 10.76993179663608470720790519061, 11.48025403259353863848869431157, 11.94879663068125396855491505755

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