Properties

Degree $4$
Conductor $13176900$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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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 − 5·7-s + 4·8-s + 3·9-s − 4·10-s − 6·12-s + 13-s − 10·14-s + 4·15-s + 5·16-s + 4·17-s + 6·18-s − 19-s − 6·20-s + 10·21-s + 3·23-s − 8·24-s + 3·25-s + 2·26-s − 4·27-s − 15·28-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 − 1.88·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.73·12-s + 0.277·13-s − 2.67·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s − 0.229·19-s − 1.34·20-s + 2.18·21-s + 0.625·23-s − 1.63·24-s + 3/5·25-s + 0.392·26-s − 0.769·27-s − 2.83·28-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: \(2\)
Selberg data: \((4,\ 13176900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 37 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 95 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 27 T + 299 T^{2} + 27 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 14 T + 146 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 17 T + 239 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−8.131396103738664002563947691408, −7.75990508175538024717433768943, −7.38031093531554931148070203961, −7.15430403376256866444549091850, −6.54602188916894816412668355635, −6.54461564772389815547244424725, −6.01230982118239416469352634948, −5.85543514909425100242439875061, −5.26944688824655288825594300211, −5.04907235577840709778785839998, −4.38449580434776504274105986292, −4.29945039865810197247479018627, −3.59920611396912255574573407326, −3.41365641417666246485100237491, −2.95154696503534740493843135233, −2.72726370297914536668279575071, −1.48036518583715417267933486074, −1.40015458791826212768109164545, 0, 0, 1.40015458791826212768109164545, 1.48036518583715417267933486074, 2.72726370297914536668279575071, 2.95154696503534740493843135233, 3.41365641417666246485100237491, 3.59920611396912255574573407326, 4.29945039865810197247479018627, 4.38449580434776504274105986292, 5.04907235577840709778785839998, 5.26944688824655288825594300211, 5.85543514909425100242439875061, 6.01230982118239416469352634948, 6.54461564772389815547244424725, 6.54602188916894816412668355635, 7.15430403376256866444549091850, 7.38031093531554931148070203961, 7.75990508175538024717433768943, 8.131396103738664002563947691408

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