Properties

Degree $4$
Conductor $78652$
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 + 3·7-s − 4·9-s − 2·11-s + 4·13-s + 16-s + 6·25-s − 3·28-s − 8·29-s + 4·36-s + 16·37-s + 6·43-s + 2·44-s − 6·47-s + 6·49-s − 4·52-s + 2·53-s + 18·59-s − 12·63-s − 64-s − 6·77-s + 7·81-s − 10·89-s + 12·91-s + 10·97-s + 8·99-s − 6·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.13·7-s − 4/3·9-s − 0.603·11-s + 1.10·13-s + 1/4·16-s + 6/5·25-s − 0.566·28-s − 1.48·29-s + 2/3·36-s + 2.63·37-s + 0.914·43-s + 0.301·44-s − 0.875·47-s + 6/7·49-s − 0.554·52-s + 0.274·53-s + 2.34·59-s − 1.51·63-s − 1/8·64-s − 0.683·77-s + 7/9·81-s − 1.05·89-s + 1.25·91-s + 1.01·97-s + 0.804·99-s − 3/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78652\)    =    \(2^{2} \cdot 7 \cdot 53^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{78652} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 78652,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.334321031\)
\(L(\frac12)\) \(\approx\) \(1.334321031\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
53$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−9.655301133683724533147904251427, −9.088948611584602373618377103382, −8.723677614040494738820191947893, −8.283365464111883779020445810788, −7.894369919682026886962231153974, −7.41957209553557856103367126899, −6.60426027611404589833447721630, −5.88137439219913863239091983650, −5.60399768289883636641912854791, −5.02228693345540321563267888365, −4.38383730034678102546747119183, −3.74040259217693484370315968302, −2.92625855173353612409306421631, −2.20126861223964646240155337573, −0.943395527170695451833846699949, 0.943395527170695451833846699949, 2.20126861223964646240155337573, 2.92625855173353612409306421631, 3.74040259217693484370315968302, 4.38383730034678102546747119183, 5.02228693345540321563267888365, 5.60399768289883636641912854791, 5.88137439219913863239091983650, 6.60426027611404589833447721630, 7.41957209553557856103367126899, 7.894369919682026886962231153974, 8.283365464111883779020445810788, 8.723677614040494738820191947893, 9.088948611584602373618377103382, 9.655301133683724533147904251427

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