Properties

Label 4-114e2-1.1-c1e2-0-2
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 6·5-s + 3·6-s + 2·7-s + 8-s + 6·9-s − 6·10-s + 9·13-s − 2·14-s − 18·15-s − 16-s − 6·17-s − 6·18-s − 8·19-s − 6·21-s − 3·24-s + 19·25-s − 9·26-s − 9·27-s − 6·29-s + 18·30-s + 6·34-s + 12·35-s + 8·38-s − 27·39-s + 6·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 2.68·5-s + 1.22·6-s + 0.755·7-s + 0.353·8-s + 2·9-s − 1.89·10-s + 2.49·13-s − 0.534·14-s − 4.64·15-s − 1/4·16-s − 1.45·17-s − 1.41·18-s − 1.83·19-s − 1.30·21-s − 0.612·24-s + 19/5·25-s − 1.76·26-s − 1.73·27-s − 1.11·29-s + 3.28·30-s + 1.02·34-s + 2.02·35-s + 1.29·38-s − 4.32·39-s + 0.948·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7178127725\)
\(L(\frac12)\) \(\approx\) \(0.7178127725\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 95 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−13.53655287403997642484764081269, −13.27434760119080765523981556636, −13.06125188585223144672285340562, −12.37072696131862652715569093396, −11.36889655736457676151856995986, −11.04861214084153788955896342041, −10.59060387862236619016136009864, −10.52385192800133151758798951739, −9.632275193483914585235363904954, −9.186213420805656112654786863573, −8.684846358385062173216292270885, −8.085225391399459615181427699381, −6.73473680322314757227157679596, −6.39482164787404369910238904729, −6.15889112416089991581995474411, −5.45950109600893872042994927731, −4.91125671332175555338440680923, −4.03354411211094901129353740907, −1.95603494038211965070339268295, −1.54712672288837368532835762924, 1.54712672288837368532835762924, 1.95603494038211965070339268295, 4.03354411211094901129353740907, 4.91125671332175555338440680923, 5.45950109600893872042994927731, 6.15889112416089991581995474411, 6.39482164787404369910238904729, 6.73473680322314757227157679596, 8.085225391399459615181427699381, 8.684846358385062173216292270885, 9.186213420805656112654786863573, 9.632275193483914585235363904954, 10.52385192800133151758798951739, 10.59060387862236619016136009864, 11.04861214084153788955896342041, 11.36889655736457676151856995986, 12.37072696131862652715569093396, 13.06125188585223144672285340562, 13.27434760119080765523981556636, 13.53655287403997642484764081269

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