Properties

Label 4-36992-1.1-c1e2-0-6
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 + 16-s + 6·17-s − 2·18-s + 8·19-s − 2·25-s + 32-s + 6·34-s − 2·36-s + 8·38-s − 8·43-s − 14·49-s − 2·50-s + 24·59-s + 64-s + 8·67-s + 6·68-s − 2·72-s + 8·76-s − 5·81-s − 24·83-s − 8·86-s − 12·89-s − 14·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 1.83·19-s − 2/5·25-s + 0.176·32-s + 1.02·34-s − 1/3·36-s + 1.29·38-s − 1.21·43-s − 2·49-s − 0.282·50-s + 3.12·59-s + 1/8·64-s + 0.977·67-s + 0.727·68-s − 0.235·72-s + 0.917·76-s − 5/9·81-s − 2.63·83-s − 0.862·86-s − 1.27·89-s − 1.41·98-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

\(L(1)\) \(\approx\) \(1.999218911\)
\(L(\frac12)\) \(\approx\) \(1.999218911\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−10.15983123501515248881660853120, −9.873162911445594563940130296972, −9.593452940951601410700456911401, −8.539147223177960804439959720606, −8.301337482966736433257870058512, −7.57953791305428441798807495308, −7.15068490322962128148785004420, −6.50013286550477748556398630858, −5.70286962395616586565109199914, −5.41195392769187495892204050810, −4.88621722151775008450458228889, −3.83340226213198750824766172133, −3.32694740405736837471862566495, −2.66497240451006162264705843189, −1.36811180866387957946526156406, 1.36811180866387957946526156406, 2.66497240451006162264705843189, 3.32694740405736837471862566495, 3.83340226213198750824766172133, 4.88621722151775008450458228889, 5.41195392769187495892204050810, 5.70286962395616586565109199914, 6.50013286550477748556398630858, 7.15068490322962128148785004420, 7.57953791305428441798807495308, 8.301337482966736433257870058512, 8.539147223177960804439959720606, 9.593452940951601410700456911401, 9.873162911445594563940130296972, 10.15983123501515248881660853120

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