↑

Properties

Label 4-390e2-1.1-c1e2-0-7
Degree $4$
Conductor $152100$
Sign $1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
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-s − 2·5-s − 6-s + 5·7-s − 8-s − 2·10-s + 3·11-s − 5·13-s + 5·14-s + 2·15-s − 16-s + 8·17-s + 5·19-s − 5·21-s + 3·22-s + 4·23-s + 24-s + 3·25-s − 5·26-s + 27-s + 4·29-s + 2·30-s − 4·31-s − 3·33-s + 8·34-s − 10·35-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.894·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s − 0.632·10-s + 0.904·11-s − 1.38·13-s + 1.33·14-s + 0.516·15-s − 1/4·16-s + 1.94·17-s + 1.14·19-s − 1.09·21-s + 0.639·22-s + 0.834·23-s + 0.204·24-s + 3/5·25-s − 0.980·26-s + 0.192·27-s + 0.742·29-s + 0.365·30-s − 0.718·31-s − 0.522·33-s + 1.37·34-s − 1.69·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024769567\)
\(L(\frac12)\) \(\approx\) \(2.024769567\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.7.af_s
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bv
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_an
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_ah
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.53.ac_ed
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2^2$ \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_acp
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$C_2^2$ \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.89.al_bg
97$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.97.a_adt
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

−11.58781777065658408086541356225, −11.49609631662168158332695502945, −10.71019029806201543855336521509, −10.41513631541863398691472256687, −9.655925829411925947797474293582, −9.383577281595628268269351009975, −8.711213230850735519341622482761, −8.017329282219072338940324245706, −7.88127892647357733272462422502, −7.37674679339913567309724890579, −6.91521316024092169532631252059, −6.17842300625523799904966333156, −5.47887409253854378312717854064, −5.12793525022361651353586049475, −4.66754358556719115491956771686, −4.43515130368606390189860790858, −3.28864928362008128579029329385, −3.20538254383081894545876644311, −1.80402139087837853590352056605, −0.982624931644370404091049194855, 0.982624931644370404091049194855, 1.80402139087837853590352056605, 3.20538254383081894545876644311, 3.28864928362008128579029329385, 4.43515130368606390189860790858, 4.66754358556719115491956771686, 5.12793525022361651353586049475, 5.47887409253854378312717854064, 6.17842300625523799904966333156, 6.91521316024092169532631252059, 7.37674679339913567309724890579, 7.88127892647357733272462422502, 8.017329282219072338940324245706, 8.711213230850735519341622482761, 9.383577281595628268269351009975, 9.655925829411925947797474293582, 10.41513631541863398691472256687, 10.71019029806201543855336521509, 11.49609631662168158332695502945, 11.58781777065658408086541356225

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