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 + 2·9-s + 4·11-s + 4·13-s + 16-s − 9·25-s + 3·28-s + 10·29-s − 2·36-s − 2·37-s − 4·44-s + 12·47-s − 4·52-s − 10·53-s + 6·59-s − 6·63-s − 64-s − 12·77-s − 5·81-s + 8·89-s − 12·91-s + 28·97-s + 8·99-s + 9·100-s + 12·107-s − 3·112-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.13·7-s + 2/3·9-s + 1.20·11-s + 1.10·13-s + 1/4·16-s − 9/5·25-s + 0.566·28-s + 1.85·29-s − 1/3·36-s − 0.328·37-s − 0.603·44-s + 1.75·47-s − 0.554·52-s − 1.37·53-s + 0.781·59-s − 0.755·63-s − 1/8·64-s − 1.36·77-s − 5/9·81-s + 0.847·89-s − 1.25·91-s + 2.84·97-s + 0.804·99-s + 9/10·100-s + 1.16·107-s − 0.283·112-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

\(L(1)\) \(\approx\) \(1.312733656\)
\(L(\frac12)\) \(\approx\) \(1.312733656\)
\(L(\frac{3}{2})\) 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$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( 1 + 10 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 11 T + p T^{2} ) \)
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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

−9.795650769586056200697367891590, −9.298384266498655105232410704257, −8.755214041893432676865445192009, −8.464662089982337084143671680117, −7.70315085303919143003861510681, −7.17442686655248981973653702440, −6.56163243959435169480328285952, −6.11878902150901686916076496674, −5.84112961209606539695942805606, −4.76799470926388909240222734202, −4.28016809453638753193840085233, −3.63464989119258862434639966821, −3.27778787093204146078093559374, −2.03803006059276837119675767080, −0.954781547726991468665431571997, 0.954781547726991468665431571997, 2.03803006059276837119675767080, 3.27778787093204146078093559374, 3.63464989119258862434639966821, 4.28016809453638753193840085233, 4.76799470926388909240222734202, 5.84112961209606539695942805606, 6.11878902150901686916076496674, 6.56163243959435169480328285952, 7.17442686655248981973653702440, 7.70315085303919143003861510681, 8.464662089982337084143671680117, 8.755214041893432676865445192009, 9.298384266498655105232410704257, 9.795650769586056200697367891590

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