Properties

Degree $4$
Conductor $1296$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 4·13-s + 16·19-s − 10·25-s − 8·31-s − 20·37-s + 16·43-s + 34·49-s + 28·61-s − 32·67-s − 20·73-s − 8·79-s − 32·91-s + 28·97-s + 40·103-s + 4·109-s − 22·121-s + 127-s + 131-s − 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.02·7-s + 1.10·13-s + 3.67·19-s − 2·25-s − 1.43·31-s − 3.28·37-s + 2.43·43-s + 34/7·49-s + 3.58·61-s − 3.90·67-s − 2.34·73-s − 0.900·79-s − 3.35·91-s + 2.84·97-s + 3.94·103-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 11.0·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1296} (1, \cdot )$
Sato-Tate group: $N(\mathrm{U}(1))$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1296,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4915286641\)
\(L(\frac12)\) \(\approx\) \(0.4915286641\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−19.74858543777126986291158021371, −18.85850998760573807432087794673, −18.85850998760573807432087794673, −17.68732582410033881966766579011, −17.68732582410033881966766579011, −16.25038600345248605266882134733, −16.25038600345248605266882134733, −15.69696813163500519905091038725, −15.69696813163500519905091038725, −13.99634105119279079569568395122, −13.99634105119279079569568395122, −13.01055982622439917603548235428, −13.01055982622439917603548235428, −11.77437667375267836950691610862, −11.77437667375267836950691610862, −10.17441103098667470227930585744, −10.17441103098667470227930585744, −9.113424945499136957715665264292, −9.113424945499136957715665264292, −7.26646731082131852272265775116, −7.26646731082131852272265775116, −5.80268955254619590131632024787, −5.80268955254619590131632024787, −3.44334336790947687892993758586, −3.44334336790947687892993758586, 3.44334336790947687892993758586, 3.44334336790947687892993758586, 5.80268955254619590131632024787, 5.80268955254619590131632024787, 7.26646731082131852272265775116, 7.26646731082131852272265775116, 9.113424945499136957715665264292, 9.113424945499136957715665264292, 10.17441103098667470227930585744, 10.17441103098667470227930585744, 11.77437667375267836950691610862, 11.77437667375267836950691610862, 13.01055982622439917603548235428, 13.01055982622439917603548235428, 13.99634105119279079569568395122, 13.99634105119279079569568395122, 15.69696813163500519905091038725, 15.69696813163500519905091038725, 16.25038600345248605266882134733, 16.25038600345248605266882134733, 17.68732582410033881966766579011, 17.68732582410033881966766579011, 18.85850998760573807432087794673, 18.85850998760573807432087794673, 19.74858543777126986291158021371

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