↑

Properties

Label 4-560e2-1.1-c1e2-0-16
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 3·9-s − 2·11-s + 8·13-s − 15-s + 6·19-s + 21-s + 3·23-s − 8·27-s − 6·29-s + 2·33-s − 35-s + 12·37-s − 8·39-s − 14·41-s + 18·43-s + 3·45-s − 6·49-s + 6·53-s − 2·55-s − 6·57-s − 10·59-s − 5·61-s − 3·63-s + 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 9-s − 0.603·11-s + 2.21·13-s − 0.258·15-s + 1.37·19-s + 0.218·21-s + 0.625·23-s − 1.53·27-s − 1.11·29-s + 0.348·33-s − 0.169·35-s + 1.97·37-s − 1.28·39-s − 2.18·41-s + 2.74·43-s + 0.447·45-s − 6/7·49-s + 0.824·53-s − 0.269·55-s − 0.794·57-s − 1.30·59-s − 0.640·61-s − 0.377·63-s + 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.863379929\)
\(L(\frac12)\) \(\approx\) \(1.863379929\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_ac
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_r
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2^2$ \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.37.am_ed
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.41.o_fb
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.43.as_gl
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_bp
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.al_cc
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.71.au_ji
73$C_2^2$ \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_aj
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_abr
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.83.ag_gt
89$C_2^2$ \( 1 + 17 T + 200 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.89.r_hs
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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

−10.81545769711213423634493337406, −10.80723463763147944051272985690, −10.11252841241293553378422087815, −9.542037616438129192655458189346, −9.394540302613449384346049926533, −9.023995278603867748844269835958, −8.087079864340594389865829294989, −7.980099964603997934496482500941, −7.40026909795747506512597950255, −6.82420590118817994838208572581, −6.35585420377380600562678516861, −5.99838216688869170657882192375, −5.33030504777728677052733581525, −5.27036316245696335354382558959, −4.25043224926279200672368264577, −3.81197437556956402118733780452, −3.33099314948671508783093821856, −2.48723732736756050385556471902, −1.55836194380666520755259668909, −0.909456746516219948911550141527, 0.909456746516219948911550141527, 1.55836194380666520755259668909, 2.48723732736756050385556471902, 3.33099314948671508783093821856, 3.81197437556956402118733780452, 4.25043224926279200672368264577, 5.27036316245696335354382558959, 5.33030504777728677052733581525, 5.99838216688869170657882192375, 6.35585420377380600562678516861, 6.82420590118817994838208572581, 7.40026909795747506512597950255, 7.980099964603997934496482500941, 8.087079864340594389865829294989, 9.023995278603867748844269835958, 9.394540302613449384346049926533, 9.542037616438129192655458189346, 10.11252841241293553378422087815, 10.80723463763147944051272985690, 10.81545769711213423634493337406

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