Properties

Label 4-36992-1.1-c1e2-0-15
Degree $4$
Conductor $36992$
Sign $-1$
Analytic cond. $2.35864$
Root an. cond. $1.23926$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 2·9-s − 8·13-s + 16-s + 2·17-s + 2·18-s − 4·19-s − 6·25-s + 8·26-s − 32-s − 2·34-s − 2·36-s + 4·38-s − 4·43-s − 2·49-s + 6·50-s − 8·52-s + 8·53-s − 20·59-s + 64-s − 4·67-s + 2·68-s + 2·72-s − 4·76-s − 5·81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.917·19-s − 6/5·25-s + 1.56·26-s − 0.176·32-s − 0.342·34-s − 1/3·36-s + 0.648·38-s − 0.609·43-s − 2/7·49-s + 0.848·50-s − 1.10·52-s + 1.09·53-s − 2.60·59-s + 1/8·64-s − 0.488·67-s + 0.242·68-s + 0.235·72-s − 0.458·76-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36992\)    =    \(2^{7} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2.35864\)
Root analytic conductor: \(1.23926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{36992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 36992,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.04816201898213158804411391639, −9.677729324561388756920590300289, −9.031999547244992859229019665711, −8.567826405181456282159198401306, −7.977291922485477404325036573972, −7.36201326366773707229211009757, −7.20889035915696995877004824232, −6.17650433034386681089394135505, −5.88063887989034217588631771396, −4.98874214236023196544129758180, −4.50421027727542554840801729526, −3.43331410730790939911193473980, −2.63736500428663363668808453369, −1.92635709516537142164401444085, 0, 1.92635709516537142164401444085, 2.63736500428663363668808453369, 3.43331410730790939911193473980, 4.50421027727542554840801729526, 4.98874214236023196544129758180, 5.88063887989034217588631771396, 6.17650433034386681089394135505, 7.20889035915696995877004824232, 7.36201326366773707229211009757, 7.977291922485477404325036573972, 8.567826405181456282159198401306, 9.031999547244992859229019665711, 9.677729324561388756920590300289, 10.04816201898213158804411391639

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