↑

Properties

Label 4-1155e2-1.1-c1e2-0-32
Degree $4$
Conductor $1334025$
Sign $1$
Analytic cond. $85.0585$
Root an. cond. $3.03689$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 2·7-s − 3·8-s + 3·9-s − 2·10-s + 2·11-s − 2·12-s − 6·13-s + 2·14-s − 4·15-s + 16-s − 3·17-s − 3·18-s − 19-s + 2·20-s + 4·21-s − 2·22-s − 7·23-s + 6·24-s + 3·25-s + 6·26-s − 4·27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.229·19-s + 0.447·20-s + 0.872·21-s − 0.426·22-s − 1.45·23-s + 1.22·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1334025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(85.0585\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1334025,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \) 2.2.b_a
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_s
17$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bg
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_bi
23$D_{4}$ \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_cc
29$D_{4}$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_bo
31$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_bu
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_co
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_de
47$D_{4}$ \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_da
53$D_{4}$ \( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.53.p_eu
59$D_{4}$ \( 1 - 17 T + 186 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.59.ar_he
61$D_{4}$ \( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_fs
67$D_{4}$ \( 1 - 22 T + 238 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.67.aw_je
71$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_ew
73$D_{4}$ \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.73.o_gw
79$D_{4}$ \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_dq
83$D_{4}$ \( 1 + 15 T + 218 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.83.p_ik
89$D_{4}$ \( 1 + 17 T + 212 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.89.r_ie
97$D_{4}$ \( 1 + 15 T + 144 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.97.p_fo
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

−9.619147841225628573576640357938, −9.513967122264208968458849986444, −8.739180470391774304001035876449, −8.693753344415080044023494634834, −7.86183836477138691924054528738, −7.36721932825129260434292650659, −7.00023991855283618727054549133, −6.62439192388111697068340986083, −6.33124672664491325866903660357, −5.79612347605071384668189312692, −5.55684075551602445238977563965, −5.05129384442759653922738385098, −4.37273487966080637528632387399, −3.96731816426904121676312942968, −3.23581817406617446204227935853, −2.45959341744008594917124935014, −2.12850837763730799097357776973, −1.42753918033326313510965428364, 0, 0, 1.42753918033326313510965428364, 2.12850837763730799097357776973, 2.45959341744008594917124935014, 3.23581817406617446204227935853, 3.96731816426904121676312942968, 4.37273487966080637528632387399, 5.05129384442759653922738385098, 5.55684075551602445238977563965, 5.79612347605071384668189312692, 6.33124672664491325866903660357, 6.62439192388111697068340986083, 7.00023991855283618727054549133, 7.36721932825129260434292650659, 7.86183836477138691924054528738, 8.693753344415080044023494634834, 8.739180470391774304001035876449, 9.513967122264208968458849986444, 9.619147841225628573576640357938

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