Properties

Label 4-147968-1.1-c1e2-0-44
Degree $4$
Conductor $147968$
Sign $-1$
Analytic cond. $9.43456$
Root an. cond. $1.75259$
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  + 2·9-s − 4·13-s − 2·17-s − 12·19-s + 6·25-s − 12·43-s − 2·49-s − 12·53-s + 4·59-s + 4·67-s − 5·81-s − 20·83-s − 4·89-s − 4·101-s + 24·103-s − 8·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2/3·9-s − 1.10·13-s − 0.485·17-s − 2.75·19-s + 6/5·25-s − 1.82·43-s − 2/7·49-s − 1.64·53-s + 0.520·59-s + 0.488·67-s − 5/9·81-s − 2.19·83-s − 0.423·89-s − 0.398·101-s + 2.36·103-s − 0.739·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147968\)    =    \(2^{9} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(9.43456\)
Root analytic conductor: \(1.75259\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 147968,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.859819645682052891450337063984, −8.671561905898679706194247095753, −8.175044239802390320790601426645, −7.57933576833918871837223986837, −6.96731719569486129906128438525, −6.63058108372294010715602043933, −6.28536542728745409427342760425, −5.45419371141816186614774462588, −4.68373951326307150779119826501, −4.59462667804277954187299746647, −3.88457241498809361151043548176, −3.02057127981915205033315412783, −2.30325689247246629597699031812, −1.63973978650798010314004838973, 0, 1.63973978650798010314004838973, 2.30325689247246629597699031812, 3.02057127981915205033315412783, 3.88457241498809361151043548176, 4.59462667804277954187299746647, 4.68373951326307150779119826501, 5.45419371141816186614774462588, 6.28536542728745409427342760425, 6.63058108372294010715602043933, 6.96731719569486129906128438525, 7.57933576833918871837223986837, 8.175044239802390320790601426645, 8.671561905898679706194247095753, 8.859819645682052891450337063984

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