Properties

Label 4-36992-1.1-c1e2-0-10
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 $0$

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 − 12·19-s − 6·25-s + 8·26-s + 32-s + 2·34-s + 2·36-s − 12·38-s + 12·43-s − 16·47-s + 2·49-s − 6·50-s + 8·52-s + 8·53-s − 4·59-s + 64-s + 4·67-s + 2·68-s + 2·72-s − 12·76-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 − 2.75·19-s − 6/5·25-s + 1.56·26-s + 0.176·32-s + 0.342·34-s + 1/3·36-s − 1.94·38-s + 1.82·43-s − 2.33·47-s + 2/7·49-s − 0.848·50-s + 1.10·52-s + 1.09·53-s − 0.520·59-s + 1/8·64-s + 0.488·67-s + 0.242·68-s + 0.235·72-s − 1.37·76-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: \(0\)
Selberg data: \((4,\ 36992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.140055600\)
\(L(\frac12)\) \(\approx\) \(2.140055600\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−10.60389981157573407446217034242, −9.966194783857914863126754522383, −9.266880437997485843191118396195, −8.712922597822970803022394832787, −8.149170853378179717399068124103, −7.83374365040642826619777615116, −6.88358390210605052670411460647, −6.39302249923793739922396430187, −6.08147959600045434700744078524, −5.43119973371138092443187760147, −4.46426130294909810305482256167, −3.99453713781203420250726150743, −3.58173899152872854729471411046, −2.37842004975190953552844472274, −1.49361093969928738645483145668, 1.49361093969928738645483145668, 2.37842004975190953552844472274, 3.58173899152872854729471411046, 3.99453713781203420250726150743, 4.46426130294909810305482256167, 5.43119973371138092443187760147, 6.08147959600045434700744078524, 6.39302249923793739922396430187, 6.88358390210605052670411460647, 7.83374365040642826619777615116, 8.149170853378179717399068124103, 8.712922597822970803022394832787, 9.266880437997485843191118396195, 9.966194783857914863126754522383, 10.60389981157573407446217034242

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