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 + 4·7-s + 4·8-s + 3·9-s + 4·10-s + 6·12-s + 6·13-s + 8·14-s + 4·15-s + 5·16-s + 4·17-s + 6·18-s − 2·19-s + 6·20-s + 8·21-s − 6·23-s + 8·24-s + 3·25-s + 12·26-s + 4·27-s + 12·28-s − 6·29-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 + 1.51·7-s + 1.41·8-s + 9-s + 1.26·10-s + 1.73·12-s + 1.66·13-s + 2.13·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s − 0.458·19-s + 1.34·20-s + 1.74·21-s − 1.25·23-s + 1.63·24-s + 3/5·25-s + 2.35·26-s + 0.769·27-s + 2.26·28-s − 1.11·29-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

\(L(1)\) \(\approx\) \(22.83562022\)
\(L(\frac12)\) \(\approx\) \(22.83562022\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 152 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 59 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 34 T + 480 T^{2} + 34 p T^{3} + p^{2} T^{4} \)
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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

−8.394268099470493082402590767464, −8.387457645867617326135688056787, −7.910554230765527998978303499899, −7.81703851237210618467699636495, −7.07747027785791675460445911131, −7.02546373954685779334712628009, −6.17863995950606854690903728613, −6.09156048186245265312117252397, −5.60901656647989014352336919055, −5.54518407433964995818255149902, −4.73722295130883785821621129594, −4.40148008582044716569074615489, −4.23729570019971350028836290756, −3.73239373445252563354893589814, −3.18675843951102880262587063785, −2.94969540625736663893497656088, −2.13428281247480610256209125136, −2.08239217263590813845521553259, −1.33019107023578876903192351829, −1.21369092461611589623152014380, 1.21369092461611589623152014380, 1.33019107023578876903192351829, 2.08239217263590813845521553259, 2.13428281247480610256209125136, 2.94969540625736663893497656088, 3.18675843951102880262587063785, 3.73239373445252563354893589814, 4.23729570019971350028836290756, 4.40148008582044716569074615489, 4.73722295130883785821621129594, 5.54518407433964995818255149902, 5.60901656647989014352336919055, 6.09156048186245265312117252397, 6.17863995950606854690903728613, 7.02546373954685779334712628009, 7.07747027785791675460445911131, 7.81703851237210618467699636495, 7.910554230765527998978303499899, 8.387457645867617326135688056787, 8.394268099470493082402590767464

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