Properties

Label 4-51e4-1.1-c1e2-0-2
Degree $4$
Conductor $6765201$
Sign $1$
Analytic cond. $431.355$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3·5-s + 3·8-s + 3·10-s − 11-s + 5·13-s + 16-s + 3·19-s + 3·20-s − 22-s − 9·23-s + 25-s + 5·26-s + 2·31-s − 32-s + 2·37-s + 3·38-s + 9·40-s − 3·41-s − 3·43-s − 44-s − 9·46-s + 14·47-s − 14·49-s + 50-s + 5·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s + 1.06·8-s + 0.948·10-s − 0.301·11-s + 1.38·13-s + 1/4·16-s + 0.688·19-s + 0.670·20-s − 0.213·22-s − 1.87·23-s + 1/5·25-s + 0.980·26-s + 0.359·31-s − 0.176·32-s + 0.328·37-s + 0.486·38-s + 1.42·40-s − 0.468·41-s − 0.457·43-s − 0.150·44-s − 1.32·46-s + 2.04·47-s − 2·49-s + 0.141·50-s + 0.693·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6765201\)    =    \(3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(431.355\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2601} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6765201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.118327335\)
\(L(\frac12)\) \(\approx\) \(6.118327335\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 \)
good2$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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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

−9.095080720216057261559829369743, −8.691007010141527775993365622587, −8.201695813013056202761873905065, −7.82508879498671445510862549510, −7.71713945820423322372321225555, −6.87914538392158510921795072217, −6.80807488236005017517235057739, −6.10283046509718605729648215108, −5.89342873739638910614173880654, −5.87257012845642446159513731338, −4.98033875699003734265981201819, −4.97877944597121344092980894332, −4.20276653614778044029884783156, −4.03559016806998891368990293116, −3.29516692515687061560349644132, −3.07199425046540054150103499234, −2.24617057566866430647133142553, −1.75679837137428861044719436053, −1.70113168832636473043419578452, −0.69940198466606463326566818198, 0.69940198466606463326566818198, 1.70113168832636473043419578452, 1.75679837137428861044719436053, 2.24617057566866430647133142553, 3.07199425046540054150103499234, 3.29516692515687061560349644132, 4.03559016806998891368990293116, 4.20276653614778044029884783156, 4.97877944597121344092980894332, 4.98033875699003734265981201819, 5.87257012845642446159513731338, 5.89342873739638910614173880654, 6.10283046509718605729648215108, 6.80807488236005017517235057739, 6.87914538392158510921795072217, 7.71713945820423322372321225555, 7.82508879498671445510862549510, 8.201695813013056202761873905065, 8.691007010141527775993365622587, 9.095080720216057261559829369743

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