Properties

Degree $4$
Conductor $4312$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·7-s − 8-s + 4·9-s − 7·11-s − 2·14-s + 16-s − 4·18-s + 7·22-s + 2·25-s + 2·28-s − 32-s + 4·36-s − 2·37-s − 8·43-s − 7·44-s − 3·49-s − 2·50-s − 6·53-s − 2·56-s + 8·63-s + 64-s − 2·67-s − 18·71-s − 4·72-s + 2·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 4/3·9-s − 2.11·11-s − 0.534·14-s + 1/4·16-s − 0.942·18-s + 1.49·22-s + 2/5·25-s + 0.377·28-s − 0.176·32-s + 2/3·36-s − 0.328·37-s − 1.21·43-s − 1.05·44-s − 3/7·49-s − 0.282·50-s − 0.824·53-s − 0.267·56-s + 1.00·63-s + 1/8·64-s − 0.244·67-s − 2.13·71-s − 0.471·72-s + 0.232·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4312\)    =    \(2^{3} \cdot 7^{2} \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6336560723\)
\(L(\frac12)\) \(\approx\) \(0.6336560723\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
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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

−12.46067757013221102019878854058, −11.72404372742540010920645685757, −11.04713528848100514340202031358, −10.52075934055572643192589332951, −10.13854250193911662745714638229, −9.584513797835638270325163339740, −8.621033754914803103925456712422, −8.141480473390208724453996247953, −7.49386781121429082057341738546, −7.10697127868267904530046410707, −6.06175654742180593464403733575, −5.10205153075762174067325072511, −4.56913088029786707885182023894, −3.12553894175809446299901750061, −1.86367941418235260420660903065, 1.86367941418235260420660903065, 3.12553894175809446299901750061, 4.56913088029786707885182023894, 5.10205153075762174067325072511, 6.06175654742180593464403733575, 7.10697127868267904530046410707, 7.49386781121429082057341738546, 8.141480473390208724453996247953, 8.621033754914803103925456712422, 9.584513797835638270325163339740, 10.13854250193911662745714638229, 10.52075934055572643192589332951, 11.04713528848100514340202031358, 11.72404372742540010920645685757, 12.46067757013221102019878854058

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