Properties

Label 4-872613-1.1-c1e2-0-1
Degree $4$
Conductor $872613$
Sign $1$
Analytic cond. $55.6385$
Root an. cond. $2.73114$
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  − 4-s + 3·7-s + 7·13-s − 3·16-s + 3·19-s + 8·25-s − 3·28-s + 4·31-s + 4·37-s − 5·43-s − 7·52-s − 11·61-s + 7·64-s + 7·67-s + 4·73-s − 3·76-s + 4·79-s + 21·91-s − 23·97-s − 8·100-s − 5·103-s + 4·109-s − 9·112-s + 2·121-s − 4·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.13·7-s + 1.94·13-s − 3/4·16-s + 0.688·19-s + 8/5·25-s − 0.566·28-s + 0.718·31-s + 0.657·37-s − 0.762·43-s − 0.970·52-s − 1.40·61-s + 7/8·64-s + 0.855·67-s + 0.468·73-s − 0.344·76-s + 0.450·79-s + 2.20·91-s − 2.33·97-s − 4/5·100-s − 0.492·103-s + 0.383·109-s − 0.850·112-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 872613 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 872613 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(872613\)    =    \(3^{8} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(55.6385\)
Root analytic conductor: \(2.73114\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{872613} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 872613,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.486002897\)
\(L(\frac12)\) \(\approx\) \(2.486002897\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−8.245691845038727425377376284574, −8.055189357856056505141834851517, −7.27818225857143003823122635105, −6.83630836171991595071415612877, −6.49988241081328010014862586896, −5.86666106654098482297502563918, −5.50273748174220440212622971507, −4.87142881210604940833701150384, −4.55564350629353767248381454801, −4.16465518440648368071174356949, −3.38775119846995653558440620648, −3.05164795292623541974859575249, −2.14958461669272356544081342978, −1.41672072622822936798286689539, −0.854020626074176840617958346778, 0.854020626074176840617958346778, 1.41672072622822936798286689539, 2.14958461669272356544081342978, 3.05164795292623541974859575249, 3.38775119846995653558440620648, 4.16465518440648368071174356949, 4.55564350629353767248381454801, 4.87142881210604940833701150384, 5.50273748174220440212622971507, 5.86666106654098482297502563918, 6.49988241081328010014862586896, 6.83630836171991595071415612877, 7.27818225857143003823122635105, 8.055189357856056505141834851517, 8.245691845038727425377376284574

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