Properties

Degree $4$
Conductor $70225$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 9-s + 4·11-s − 6·13-s − 3·16-s − 10·17-s + 25-s − 2·29-s − 36-s − 2·37-s − 4·43-s − 4·44-s − 8·47-s + 2·49-s + 6·52-s − 6·53-s − 4·59-s + 7·64-s + 10·68-s − 8·81-s + 4·89-s − 6·97-s + 4·99-s − 100-s + 4·107-s + 18·113-s + 2·116-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 3/4·16-s − 2.42·17-s + 1/5·25-s − 0.371·29-s − 1/6·36-s − 0.328·37-s − 0.609·43-s − 0.603·44-s − 1.16·47-s + 2/7·49-s + 0.832·52-s − 0.824·53-s − 0.520·59-s + 7/8·64-s + 1.21·68-s − 8/9·81-s + 0.423·89-s − 0.609·97-s + 0.402·99-s − 0.0999·100-s + 0.386·107-s + 1.69·113-s + 0.185·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(70225\)    =    \(5^{2} \cdot 53^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{70225} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 70225,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
53$C_2$ \( 1 + 6 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 139 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.444343480890062656604441750970, −9.261689275474172088333610453835, −8.581524401474406523317120461660, −8.320024732246652796417716600079, −7.22451846381919744814935289051, −7.14228454788994304190179074405, −6.54758816478394080842782118150, −6.05553805608552501829032549125, −5.01837782869741371042668313406, −4.63625097959755262862409811200, −4.31271900189114347524885586539, −3.47239140183962337096373868503, −2.45897901494353778922882417762, −1.80260202495615351139380593868, 0, 1.80260202495615351139380593868, 2.45897901494353778922882417762, 3.47239140183962337096373868503, 4.31271900189114347524885586539, 4.63625097959755262862409811200, 5.01837782869741371042668313406, 6.05553805608552501829032549125, 6.54758816478394080842782118150, 7.14228454788994304190179074405, 7.22451846381919744814935289051, 8.320024732246652796417716600079, 8.581524401474406523317120461660, 9.261689275474172088333610453835, 9.444343480890062656604441750970

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