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Properties

Label 4-1155e2-1.1-c1e2-0-8
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 2·4-s − 2·5-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s − 4·10-s − 2·11-s − 4·12-s + 2·13-s + 4·14-s + 4·15-s + 8·16-s − 6·17-s + 6·18-s − 10·19-s − 4·20-s − 4·21-s − 4·22-s + 10·23-s − 8·24-s + 3·25-s + 4·26-s − 4·27-s + 4·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s − 1.15·12-s + 0.554·13-s + 1.06·14-s + 1.03·15-s + 2·16-s − 1.45·17-s + 1.41·18-s − 2.29·19-s − 0.894·20-s − 0.872·21-s − 0.852·22-s + 2.08·23-s − 1.63·24-s + 3/5·25-s + 0.784·26-s − 0.769·27-s + 0.755·28-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: \(0\)
Selberg data: \((4,\ 1334025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.597359637\)
\(L(\frac12)\) \(\approx\) \(2.597359637\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_c
13$D_{4}$ \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_a
17$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bf
19$D_{4}$ \( 1 + 10 T + 51 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.19.k_bz
23$D_{4}$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.23.ak_ch
29$D_{4}$ \( 1 - 8 T + 71 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_ct
31$D_{4}$ \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.31.o_ee
37$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_bw
41$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_i
43$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.43.a_df
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_dw
53$D_{4}$ \( 1 - 2 T + 95 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dr
59$D_{4}$ \( 1 - 24 T + 259 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.59.ay_jz
61$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_p
67$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_dm
71$D_{4}$ \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_cq
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_bq
79$D_{4}$ \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_ee
83$D_{4}$ \( 1 + 2 T + 155 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_fz
89$D_{4}$ \( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_fj
97$D_{4}$ \( 1 + 8 T + 207 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hz
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.53058924060965135279181518257, −9.790559447414366264959554886816, −8.884757785637890613151656278668, −8.772727463393138777089024584474, −8.353636075367501471609397000585, −7.79463769549596060507922757008, −7.27286628794809562431611589932, −6.91388055414940181746465443756, −6.76136701513419556854636607380, −6.06736263841106148280350070091, −5.60135403566506504430228925351, −5.11988230490427366285750778897, −4.89727294568226387950011100045, −4.35677293668797325777684366300, −4.07479067432813116528226032217, −3.81793811967262017526755387365, −2.84823814425087349508888438448, −2.24000732852800723036843551665, −1.55077423004073157023590535648, −0.61741691731344019653415010676, 0.61741691731344019653415010676, 1.55077423004073157023590535648, 2.24000732852800723036843551665, 2.84823814425087349508888438448, 3.81793811967262017526755387365, 4.07479067432813116528226032217, 4.35677293668797325777684366300, 4.89727294568226387950011100045, 5.11988230490427366285750778897, 5.60135403566506504430228925351, 6.06736263841106148280350070091, 6.76136701513419556854636607380, 6.91388055414940181746465443756, 7.27286628794809562431611589932, 7.79463769549596060507922757008, 8.353636075367501471609397000585, 8.772727463393138777089024584474, 8.884757785637890613151656278668, 9.790559447414366264959554886816, 10.53058924060965135279181518257

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