Properties

Label 4-13132-1.1-c1e2-0-1
Degree $4$
Conductor $13132$
Sign $1$
Analytic cond. $0.837307$
Root an. cond. $0.956579$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 3·7-s + 4·8-s − 9-s + 2·11-s − 6·14-s + 5·16-s − 2·18-s + 4·22-s − 7·23-s + 4·25-s − 9·28-s − 6·29-s + 6·32-s − 3·36-s − 12·37-s + 11·43-s + 6·44-s − 14·46-s + 2·49-s + 8·50-s − 2·53-s − 12·56-s − 12·58-s + 3·63-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.13·7-s + 1.41·8-s − 1/3·9-s + 0.603·11-s − 1.60·14-s + 5/4·16-s − 0.471·18-s + 0.852·22-s − 1.45·23-s + 4/5·25-s − 1.70·28-s − 1.11·29-s + 1.06·32-s − 1/2·36-s − 1.97·37-s + 1.67·43-s + 0.904·44-s − 2.06·46-s + 2/7·49-s + 1.13·50-s − 0.274·53-s − 1.60·56-s − 1.57·58-s + 0.377·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13132\)    =    \(2^{2} \cdot 7^{2} \cdot 67\)
Sign: $1$
Analytic conductor: \(0.837307\)
Root analytic conductor: \(0.956579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13132} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13132,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.088129922\)
\(L(\frac12)\) \(\approx\) \(2.088129922\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
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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

−11.46608213493968626727300163207, −10.82760088508319938062447305558, −10.33657435048486602488996587889, −9.695683921864892940047753506823, −9.112125576411928642769877559337, −8.439644640945821413931531157457, −7.57452056108471410921263826525, −7.01219029556816612797314956669, −6.41820765993181121572351111656, −5.90511758387739515136791413173, −5.34383943955583570994254221153, −4.40308216215535518235923432343, −3.70870276908503450949320302427, −3.15624900586311890679207826246, −2.06411574815739123744459976512, 2.06411574815739123744459976512, 3.15624900586311890679207826246, 3.70870276908503450949320302427, 4.40308216215535518235923432343, 5.34383943955583570994254221153, 5.90511758387739515136791413173, 6.41820765993181121572351111656, 7.01219029556816612797314956669, 7.57452056108471410921263826525, 8.439644640945821413931531157457, 9.112125576411928642769877559337, 9.695683921864892940047753506823, 10.33657435048486602488996587889, 10.82760088508319938062447305558, 11.46608213493968626727300163207

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