Properties

Label 4-345600-1.1-c1e2-0-30
Degree $4$
Conductor $345600$
Sign $-1$
Analytic cond. $22.0357$
Root an. cond. $2.16661$
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  + 3-s − 2·5-s + 9-s − 2·15-s − 8·19-s + 3·25-s + 27-s + 4·29-s + 24·43-s − 2·45-s − 16·47-s − 14·49-s − 12·53-s − 8·57-s + 8·67-s − 16·71-s − 12·73-s + 3·75-s + 81-s + 4·87-s + 16·95-s + 4·97-s − 12·101-s − 6·121-s − 4·125-s + 127-s + 24·129-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.516·15-s − 1.83·19-s + 3/5·25-s + 0.192·27-s + 0.742·29-s + 3.65·43-s − 0.298·45-s − 2.33·47-s − 2·49-s − 1.64·53-s − 1.05·57-s + 0.977·67-s − 1.89·71-s − 1.40·73-s + 0.346·75-s + 1/9·81-s + 0.428·87-s + 1.64·95-s + 0.406·97-s − 1.19·101-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 2.11·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345600\)    =    \(2^{9} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(22.0357\)
Root analytic conductor: \(2.16661\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 345600,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.645810177900386367635479957201, −7.982386705736041440880055703756, −7.76560916456884806442341251241, −7.23083008412003813869292936859, −6.58285993178476791834968599347, −6.33640038439988253692354509575, −5.74168392413099570963868387782, −4.88793662384328778723790668153, −4.41954513526927827595737015950, −4.19803603229033482796144552370, −3.38984665882479408513314051846, −2.92311775669320789657547583777, −2.22526038506206736624914155682, −1.36054514292875137606341288502, 0, 1.36054514292875137606341288502, 2.22526038506206736624914155682, 2.92311775669320789657547583777, 3.38984665882479408513314051846, 4.19803603229033482796144552370, 4.41954513526927827595737015950, 4.88793662384328778723790668153, 5.74168392413099570963868387782, 6.33640038439988253692354509575, 6.58285993178476791834968599347, 7.23083008412003813869292936859, 7.76560916456884806442341251241, 7.982386705736041440880055703756, 8.645810177900386367635479957201

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