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Properties

Label 4-405e2-1.1-c1e2-0-8
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $10.4583$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 5-s + 3·7-s + 4·8-s + 2·10-s + 2·11-s + 5·13-s + 6·14-s + 8·16-s − 16·17-s + 2·19-s + 2·20-s + 4·22-s − 6·23-s + 10·26-s + 6·28-s − 2·29-s + 8·32-s − 32·34-s + 3·35-s + 10·37-s + 4·38-s + 4·40-s + 10·41-s − 4·43-s + 4·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s + 1.13·7-s + 1.41·8-s + 0.632·10-s + 0.603·11-s + 1.38·13-s + 1.60·14-s + 2·16-s − 3.88·17-s + 0.458·19-s + 0.447·20-s + 0.852·22-s − 1.25·23-s + 1.96·26-s + 1.13·28-s − 0.371·29-s + 1.41·32-s − 5.48·34-s + 0.507·35-s + 1.64·37-s + 0.648·38-s + 0.632·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10.4583\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 164025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.766390820\)
\(L(\frac12)\) \(\approx\) \(4.766390820\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_c
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_c
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_du
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_az
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_ch
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_abb
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_abf
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_f
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.67.aj_o
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.73.k_gp
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_acs
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_abv
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−11.64645548910393506153558759244, −11.22882763042049182602553103669, −10.70322080651256461515156567249, −10.55245329301344860005968789240, −9.484172136300199741271880591972, −9.369790042733689608550413883129, −8.489921626606955614305488679115, −8.389656835759033156851899294891, −7.73910683689901688485496081239, −7.09209990093959457095039561185, −6.57678207626383116409346854508, −6.17554562770879025472472323201, −5.71037283355038685346066876541, −5.03289313192704901915783090092, −4.46209447012997208372924027680, −4.13110739194272935515026937616, −3.93138456830584759827346734526, −2.66895190097342901208957581953, −2.02672669427142178442363967082, −1.43317395717623672860292060019, 1.43317395717623672860292060019, 2.02672669427142178442363967082, 2.66895190097342901208957581953, 3.93138456830584759827346734526, 4.13110739194272935515026937616, 4.46209447012997208372924027680, 5.03289313192704901915783090092, 5.71037283355038685346066876541, 6.17554562770879025472472323201, 6.57678207626383116409346854508, 7.09209990093959457095039561185, 7.73910683689901688485496081239, 8.389656835759033156851899294891, 8.489921626606955614305488679115, 9.369790042733689608550413883129, 9.484172136300199741271880591972, 10.55245329301344860005968789240, 10.70322080651256461515156567249, 11.22882763042049182602553103669, 11.64645548910393506153558759244

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