↑

Properties

Label 4-735e2-1.1-c1e2-0-11
Degree $4$
Conductor $540225$
Sign $1$
Analytic cond. $34.4452$
Root an. cond. $2.42260$
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  − 3·2-s + 3·3-s + 4·4-s + 5-s − 9·6-s − 3·8-s + 6·9-s − 3·10-s − 6·11-s + 12·12-s + 3·15-s + 3·16-s + 6·17-s − 18·18-s + 12·19-s + 4·20-s + 18·22-s + 3·23-s − 9·24-s + 9·27-s − 9·30-s + 6·31-s − 6·32-s − 18·33-s − 18·34-s + 24·36-s − 4·37-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 2·4-s + 0.447·5-s − 3.67·6-s − 1.06·8-s + 2·9-s − 0.948·10-s − 1.80·11-s + 3.46·12-s + 0.774·15-s + 3/4·16-s + 1.45·17-s − 4.24·18-s + 2.75·19-s + 0.894·20-s + 3.83·22-s + 0.625·23-s − 1.83·24-s + 1.73·27-s − 1.64·30-s + 1.07·31-s − 1.06·32-s − 3.13·33-s − 3.08·34-s + 4·36-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(34.4452\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 540225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.372496600\)
\(L(\frac12)\) \(\approx\) \(1.372496600\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_x
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.19.am_cp
23$C_2^2$ \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ba
29$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.29.a_acd
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_br
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_av
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_dk
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_dy
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2^2$ \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_dh
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_gv
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_adc
97$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.97.a_adi
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

−10.12081795069356890849177740334, −9.861560244749447161958471116739, −9.762461707724518817279864473853, −9.380639126804241796461073885061, −8.791420692307502560887036280968, −8.652754806225204056131835372910, −7.958168204422630260044856493039, −7.70827584720661342318085965207, −7.66754596454278626410245235211, −7.22406830593214149349290596604, −6.47324495057860136401268846122, −5.70352292515603276330337282614, −5.13490071347792168220424589409, −4.96698557208501029033237445609, −3.52943232193821911354473291061, −3.52029523256780488347456354629, −2.58483747134084359262287939473, −2.49639712592843894223475676327, −1.30571291054086267471221576766, −0.915060577942144369755245303036, 0.915060577942144369755245303036, 1.30571291054086267471221576766, 2.49639712592843894223475676327, 2.58483747134084359262287939473, 3.52029523256780488347456354629, 3.52943232193821911354473291061, 4.96698557208501029033237445609, 5.13490071347792168220424589409, 5.70352292515603276330337282614, 6.47324495057860136401268846122, 7.22406830593214149349290596604, 7.66754596454278626410245235211, 7.70827584720661342318085965207, 7.958168204422630260044856493039, 8.652754806225204056131835372910, 8.791420692307502560887036280968, 9.380639126804241796461073885061, 9.762461707724518817279864473853, 9.861560244749447161958471116739, 10.12081795069356890849177740334

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