Properties

Label 4-18e4-1.1-c1e2-0-4
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $6.69336$
Root an. cond. $1.60846$
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  + 4·7-s − 2·13-s + 16·19-s + 5·25-s + 4·31-s − 20·37-s − 8·43-s + 7·49-s − 14·61-s + 16·67-s − 20·73-s + 4·79-s − 8·91-s − 14·97-s − 20·103-s + 4·109-s + 11·121-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.554·13-s + 3.67·19-s + 25-s + 0.718·31-s − 3.28·37-s − 1.21·43-s + 49-s − 1.79·61-s + 1.95·67-s − 2.34·73-s + 0.450·79-s − 0.838·91-s − 1.42·97-s − 1.97·103-s + 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.69336\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{324} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.966114656\)
\(L(\frac12)\) \(\approx\) \(1.966114656\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−11.82235422542429105925070832049, −11.46958622415999571144457931992, −10.98076526105920228072948946830, −10.51337003153214371961132201644, −9.892946035630855793755962860148, −9.671473231001205570019232826521, −8.961855264884587509955799701775, −8.557297404849904883139388388967, −7.997507265526095423109297164313, −7.61686246832907793041088019723, −6.95141502013665706448618407516, −6.91031273329751123854354002521, −5.58855634823398388109491201333, −5.39649456725599843144128363972, −4.92950719856807186528671678599, −4.44450720019173676674692811510, −3.24801769331644665393840249736, −3.14422391153317343904853441402, −1.83735858653833911855574313271, −1.18259588849665885008193160449, 1.18259588849665885008193160449, 1.83735858653833911855574313271, 3.14422391153317343904853441402, 3.24801769331644665393840249736, 4.44450720019173676674692811510, 4.92950719856807186528671678599, 5.39649456725599843144128363972, 5.58855634823398388109491201333, 6.91031273329751123854354002521, 6.95141502013665706448618407516, 7.61686246832907793041088019723, 7.997507265526095423109297164313, 8.557297404849904883139388388967, 8.961855264884587509955799701775, 9.671473231001205570019232826521, 9.892946035630855793755962860148, 10.51337003153214371961132201644, 10.98076526105920228072948946830, 11.46958622415999571144457931992, 11.82235422542429105925070832049

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