↑

Properties

Label 4-1707552-1.1-c1e2-0-32
Degree $4$
Conductor $1707552$
Sign $-1$
Analytic cond. $108.874$
Root an. cond. $3.23021$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·7-s − 8-s + 9-s + 3·11-s + 2·14-s + 16-s − 18-s − 3·22-s − 2·23-s + 2·25-s − 2·28-s − 29-s − 32-s + 36-s + 5·37-s + 43-s + 3·44-s + 2·46-s − 3·49-s − 2·50-s − 7·53-s + 2·56-s + 58-s − 2·63-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.534·14-s + 1/4·16-s − 0.235·18-s − 0.639·22-s − 0.417·23-s + 2/5·25-s − 0.377·28-s − 0.185·29-s − 0.176·32-s + 1/6·36-s + 0.821·37-s + 0.152·43-s + 0.452·44-s + 0.294·46-s − 3/7·49-s − 0.282·50-s − 0.961·53-s + 0.267·56-s + 0.131·58-s − 0.251·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1707552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1707552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1707552\)    =    \(2^{5} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(108.874\)
Root analytic conductor: \(3.23021\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1707552,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 - 3 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
13$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.13.a_m
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.19.a_bc
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.b_cg
31$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.31.a_ar
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.af_da
41$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \) 2.41.a_acz
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.ab_dg
47$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.47.a_br
53$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.h_eo
59$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \) 2.59.a_do
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.61.a_bo
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.m_gk
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.s_gs
73$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \) 2.73.a_dd
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.ai_fi
83$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.83.a_abc
89$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.89.a_cj
97$C_2^2$ \( 1 - 149 T^{2} + p^{2} T^{4} \) 2.97.a_aft
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.72426935857317195641995794120, −7.25194908001777634105498174257, −6.70817364790378194684152565421, −6.47852299858105369932704305830, −6.03109206039042596986967682968, −5.67963685856146347272054013618, −4.95869588364017305169212103434, −4.41832612752334635779782163226, −4.05990786741313211718330433522, −3.32581303835425683052326387771, −3.07266529097166901354371432444, −2.31872611325556599436506803523, −1.65102465034092613246001194236, −1.03728279270502598206215244314, 0, 1.03728279270502598206215244314, 1.65102465034092613246001194236, 2.31872611325556599436506803523, 3.07266529097166901354371432444, 3.32581303835425683052326387771, 4.05990786741313211718330433522, 4.41832612752334635779782163226, 4.95869588364017305169212103434, 5.67963685856146347272054013618, 6.03109206039042596986967682968, 6.47852299858105369932704305830, 6.70817364790378194684152565421, 7.25194908001777634105498174257, 7.72426935857317195641995794120

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