Properties

Degree $4$
Conductor $13176900$
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  + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s − 6·12-s + 9·13-s − 4·14-s + 4·15-s + 5·16-s − 17-s + 6·18-s − 2·19-s − 6·20-s + 4·21-s + 7·23-s − 8·24-s + 3·25-s + 18·26-s − 4·27-s − 6·28-s − 3·29-s + 8·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.73·12-s + 2.49·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s − 0.458·19-s − 1.34·20-s + 0.872·21-s + 1.45·23-s − 1.63·24-s + 3/5·25-s + 3.53·26-s − 0.769·27-s − 1.13·28-s − 0.557·29-s + 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13176900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3630} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13176900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.599668694\)
\(L(\frac12)\) \(\approx\) \(4.599668694\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 15 T + 129 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 11 T + 85 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 17 T + 159 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 129 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T - 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.663664574857548568501955385403, −8.432558793556105026022099598104, −7.67493544865306120158796026037, −7.50141446340011959048795912227, −6.98464193618107567478892230879, −6.72483259520958704336880896900, −6.46283570619403572495136915072, −6.07473107019606234201159244709, −5.60580644063062789793460646558, −5.50173183784178937384952316295, −4.78583037474975579078024395617, −4.74833463073566246059108095322, −3.90004228016573461707863764169, −3.80183146455483967126646690612, −3.35193610237345354544003971592, −3.32614083794030071709771394504, −2.16509272387993420225446256538, −1.96234430021605036967134723404, −0.932696976675516260579889163180, −0.69559860921755579442895495884, 0.69559860921755579442895495884, 0.932696976675516260579889163180, 1.96234430021605036967134723404, 2.16509272387993420225446256538, 3.32614083794030071709771394504, 3.35193610237345354544003971592, 3.80183146455483967126646690612, 3.90004228016573461707863764169, 4.74833463073566246059108095322, 4.78583037474975579078024395617, 5.50173183784178937384952316295, 5.60580644063062789793460646558, 6.07473107019606234201159244709, 6.46283570619403572495136915072, 6.72483259520958704336880896900, 6.98464193618107567478892230879, 7.50141446340011959048795912227, 7.67493544865306120158796026037, 8.432558793556105026022099598104, 8.663664574857548568501955385403

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