Properties

Degree $4$
Conductor $103933$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·7-s − 9-s + 11-s + 2·13-s − 3·16-s + 11·17-s + 2·25-s + 2·28-s + 4·29-s − 36-s − 6·37-s − 13·43-s + 44-s + 2·47-s − 7·49-s + 2·52-s + 12·53-s − 7·59-s − 2·63-s − 7·64-s + 11·68-s + 2·77-s − 8·81-s + 7·89-s + 4·91-s + 11·97-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s − 1/3·9-s + 0.301·11-s + 0.554·13-s − 3/4·16-s + 2.66·17-s + 2/5·25-s + 0.377·28-s + 0.742·29-s − 1/6·36-s − 0.986·37-s − 1.98·43-s + 0.150·44-s + 0.291·47-s − 49-s + 0.277·52-s + 1.64·53-s − 0.911·59-s − 0.251·63-s − 7/8·64-s + 1.33·68-s + 0.227·77-s − 8/9·81-s + 0.741·89-s + 0.419·91-s + 1.11·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103933 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103933 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(103933\)    =    \(37 \cdot 53^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{103933} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 103933,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041648123\)
\(L(\frac12)\) \(\approx\) \(2.041648123\)
\(L(\frac{3}{2})\) 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)$
bad37$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
53$C_2$ \( 1 - 12 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 67 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−9.605235318577818749207375680876, −8.941542101630233843686553287236, −8.507028364869637329472577925242, −8.091771473777809654936138879841, −7.59937666342427318290468930780, −7.06953373283624440544238162175, −6.52812101801995199603237348885, −5.99834956749977380559884209127, −5.33049792011248494870731111904, −5.01968553020813205942679036578, −4.21170458235797444211830897896, −3.38139265046398272718151717174, −3.02419162119377151430180875075, −1.91756666525836360224607595102, −1.18273373633220554792050098352, 1.18273373633220554792050098352, 1.91756666525836360224607595102, 3.02419162119377151430180875075, 3.38139265046398272718151717174, 4.21170458235797444211830897896, 5.01968553020813205942679036578, 5.33049792011248494870731111904, 5.99834956749977380559884209127, 6.52812101801995199603237348885, 7.06953373283624440544238162175, 7.59937666342427318290468930780, 8.091771473777809654936138879841, 8.507028364869637329472577925242, 8.941542101630233843686553287236, 9.605235318577818749207375680876

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