Properties

Label 4-128772-1.1-c1e2-0-0
Degree $4$
Conductor $128772$
Sign $1$
Analytic cond. $8.21061$
Root an. cond. $1.69275$
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  + 2·3-s − 4-s + 9-s − 2·12-s + 10·13-s + 16-s + 8·19-s − 6·25-s − 4·27-s − 36-s + 20·39-s − 6·43-s + 2·48-s − 7·49-s − 10·52-s + 16·57-s + 4·61-s − 64-s + 73-s − 12·75-s − 8·76-s − 4·79-s − 11·81-s + 12·97-s + 6·100-s + 20·103-s + 4·108-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 1/3·9-s − 0.577·12-s + 2.77·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s − 0.769·27-s − 1/6·36-s + 3.20·39-s − 0.914·43-s + 0.288·48-s − 49-s − 1.38·52-s + 2.11·57-s + 0.512·61-s − 1/8·64-s + 0.117·73-s − 1.38·75-s − 0.917·76-s − 0.450·79-s − 1.22·81-s + 1.21·97-s + 3/5·100-s + 1.97·103-s + 0.384·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(128772\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73\)
Sign: $1$
Analytic conductor: \(8.21061\)
Root analytic conductor: \(1.69275\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{128772} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 128772,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.378728304\)
\(L(\frac12)\) \(\approx\) \(2.378728304\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.467032885365556502067082448345, −8.782354457074067788667432274050, −8.454937496721905083310116276290, −8.111677634284937459268485258182, −7.65838128118712585772280068505, −7.05312709769497740884594959130, −6.30715776902446325868057355369, −5.85277217746542879607550620363, −5.41888271791723788556051307459, −4.59430410867661782446311846336, −3.81959466783682612274917457509, −3.47610698181349043818130621017, −3.10459274821384724745702027716, −1.93963279963835347501089062311, −1.15432602697019122561248537204, 1.15432602697019122561248537204, 1.93963279963835347501089062311, 3.10459274821384724745702027716, 3.47610698181349043818130621017, 3.81959466783682612274917457509, 4.59430410867661782446311846336, 5.41888271791723788556051307459, 5.85277217746542879607550620363, 6.30715776902446325868057355369, 7.05312709769497740884594959130, 7.65838128118712585772280068505, 8.111677634284937459268485258182, 8.454937496721905083310116276290, 8.782354457074067788667432274050, 9.467032885365556502067082448345

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