Properties

Degree $4$
Conductor $12446784$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·11-s + 4·13-s − 6·17-s + 4·19-s − 6·23-s + 5·25-s + 4·31-s − 10·37-s + 4·41-s − 8·43-s − 4·47-s + 12·53-s + 4·55-s − 12·59-s − 6·61-s − 8·65-s + 4·67-s − 28·71-s + 2·73-s + 8·79-s − 32·83-s + 12·85-s + 6·89-s − 8·95-s − 36·97-s − 14·101-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.603·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 25-s + 0.718·31-s − 1.64·37-s + 0.624·41-s − 1.21·43-s − 0.583·47-s + 1.64·53-s + 0.539·55-s − 1.56·59-s − 0.768·61-s − 0.992·65-s + 0.488·67-s − 3.32·71-s + 0.234·73-s + 0.900·79-s − 3.51·83-s + 1.30·85-s + 0.635·89-s − 0.820·95-s − 3.65·97-s − 1.39·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1030231916\)
\(L(\frac12)\) \(\approx\) \(0.1030231916\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−8.890078517589631724907701632547, −8.356650412532347210833203915672, −8.165833121720289315957894878189, −7.64321550013826848539078481267, −7.29137050779089477964824882642, −6.89375100602734023606623402807, −6.68304848926556468219948100748, −5.98356369611582582772717245715, −5.96601860841122728004362357105, −5.14271294802645527691279459149, −5.13654557958103102800689401193, −4.31037158172445040616077874153, −4.21461812439350203914778916789, −3.80227330923544098251750680524, −3.18771686157930484868623176428, −2.85694728215589914817387002134, −2.42113287979179225495843421819, −1.47014701053706603693312788959, −1.38496393885218608056041179801, −0.097056364725263414226705901230, 0.097056364725263414226705901230, 1.38496393885218608056041179801, 1.47014701053706603693312788959, 2.42113287979179225495843421819, 2.85694728215589914817387002134, 3.18771686157930484868623176428, 3.80227330923544098251750680524, 4.21461812439350203914778916789, 4.31037158172445040616077874153, 5.13654557958103102800689401193, 5.14271294802645527691279459149, 5.96601860841122728004362357105, 5.98356369611582582772717245715, 6.68304848926556468219948100748, 6.89375100602734023606623402807, 7.29137050779089477964824882642, 7.64321550013826848539078481267, 8.165833121720289315957894878189, 8.356650412532347210833203915672, 8.890078517589631724907701632547

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