Properties

Label 4-162e2-1.1-c1e2-0-4
Degree $4$
Conductor $26244$
Sign $1$
Analytic cond. $1.67334$
Root an. cond. $1.13735$
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-s + 4·7-s + 4·13-s + 16-s − 2·19-s − 10·25-s + 4·28-s − 8·31-s − 8·37-s − 2·43-s − 2·49-s + 4·52-s + 16·61-s + 64-s + 10·67-s + 22·73-s − 2·76-s − 8·79-s + 16·91-s + 10·97-s − 10·100-s + 28·103-s − 32·109-s + 4·112-s − 13·121-s − 8·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.51·7-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 2·25-s + 0.755·28-s − 1.43·31-s − 1.31·37-s − 0.304·43-s − 2/7·49-s + 0.554·52-s + 2.04·61-s + 1/8·64-s + 1.22·67-s + 2.57·73-s − 0.229·76-s − 0.900·79-s + 1.67·91-s + 1.01·97-s − 100-s + 2.75·103-s − 3.06·109-s + 0.377·112-s − 1.18·121-s − 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26244\)    =    \(2^{2} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.67334\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{26244} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.572084960\)
\(L(\frac12)\) \(\approx\) \(1.572084960\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−10.96157598355786499811507828582, −10.11405722912123694469504278059, −9.711604924732818951217103858923, −8.789487803962222669402730529275, −8.543319215757720981308384286559, −7.80670854475898716623947697687, −7.63671497466111343831458484773, −6.68619713403218599725876916726, −6.25936593777468405983226275094, −5.34383976963704470462884839275, −5.13805326110939411578475614907, −3.94847330019587176084970982031, −3.65076060485342096132488694617, −2.19879385611801234222671009191, −1.60693251106342657124988453680, 1.60693251106342657124988453680, 2.19879385611801234222671009191, 3.65076060485342096132488694617, 3.94847330019587176084970982031, 5.13805326110939411578475614907, 5.34383976963704470462884839275, 6.25936593777468405983226275094, 6.68619713403218599725876916726, 7.63671497466111343831458484773, 7.80670854475898716623947697687, 8.543319215757720981308384286559, 8.789487803962222669402730529275, 9.711604924732818951217103858923, 10.11405722912123694469504278059, 10.96157598355786499811507828582

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