↑

Properties

Label 4-450e2-1.1-c1e2-0-3
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
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  − 2-s + 3·3-s − 3·6-s − 4·7-s + 8-s + 6·9-s − 3·11-s − 4·13-s + 4·14-s − 16-s − 6·17-s − 6·18-s + 10·19-s − 12·21-s + 3·22-s + 6·23-s + 3·24-s + 4·26-s + 9·27-s − 6·29-s − 2·31-s − 9·33-s + 6·34-s + 8·37-s − 10·38-s − 12·39-s + 3·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 1.22·6-s − 1.51·7-s + 0.353·8-s + 2·9-s − 0.904·11-s − 1.10·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s − 1.41·18-s + 2.29·19-s − 2.61·21-s + 0.639·22-s + 1.25·23-s + 0.612·24-s + 0.784·26-s + 1.73·27-s − 1.11·29-s − 0.359·31-s − 1.56·33-s + 1.02·34-s + 1.31·37-s − 1.62·38-s − 1.92·39-s + 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.500586383\)
\(L(\frac12)\) \(\approx\) \(1.500586383\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_d
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_abb
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.43.al_da
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.97.al_y
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

−11.25426202731734570921206042314, −10.68412491267172043846114535576, −10.08330039334786501292047065608, −9.683081335978849481986651908904, −9.464320649240525123403899053584, −9.155914723210747693732836297598, −8.901125137663368093442454341648, −8.004119542019057367218998310321, −7.80938238875230284249442390498, −7.24981100197392675649202575334, −7.10885807291669177787233778563, −6.41386011160327488620348105936, −5.63887455675272543450732233758, −4.97463179060482432892824328229, −4.48024658050051536019334713005, −3.49952481049294439147318041166, −3.34742396903206875561145136961, −2.46536595023597651361376897480, −2.29086027744572191367461993429, −0.77891632088214893603774961727, 0.77891632088214893603774961727, 2.29086027744572191367461993429, 2.46536595023597651361376897480, 3.34742396903206875561145136961, 3.49952481049294439147318041166, 4.48024658050051536019334713005, 4.97463179060482432892824328229, 5.63887455675272543450732233758, 6.41386011160327488620348105936, 7.10885807291669177787233778563, 7.24981100197392675649202575334, 7.80938238875230284249442390498, 8.004119542019057367218998310321, 8.901125137663368093442454341648, 9.155914723210747693732836297598, 9.464320649240525123403899053584, 9.683081335978849481986651908904, 10.08330039334786501292047065608, 10.68412491267172043846114535576, 11.25426202731734570921206042314

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