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Properties

Label 4-405e2-1.1-c1e2-0-2
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 − 2·5-s − 6·7-s + 4·8-s − 4·10-s + 8·11-s − 4·13-s − 12·14-s + 8·16-s + 2·17-s + 2·19-s − 4·20-s + 16·22-s + 3·25-s − 8·26-s − 12·28-s + 4·29-s − 6·31-s + 8·32-s + 4·34-s + 12·35-s − 2·37-s + 4·38-s − 8·40-s + 4·41-s − 10·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.894·5-s − 2.26·7-s + 1.41·8-s − 1.26·10-s + 2.41·11-s − 1.10·13-s − 3.20·14-s + 2·16-s + 0.485·17-s + 0.458·19-s − 0.894·20-s + 3.41·22-s + 3/5·25-s − 1.56·26-s − 2.26·28-s + 0.742·29-s − 1.07·31-s + 1.41·32-s + 0.685·34-s + 2.02·35-s − 0.328·37-s + 0.648·38-s − 1.26·40-s + 0.624·41-s − 1.52·43-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

\(L(1)\) \(\approx\) \(2.751552543\)
\(L(\frac12)\) \(\approx\) \(2.751552543\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_c
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_u
11$D_{4}$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.11.ai_bj
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_s
17$D_{4}$ \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_bg
19$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bb
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_bj
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$D_{4}$ \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cu
41$D_{4}$ \( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_ch
43$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dg
47$D_{4}$ \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_fk
53$D_{4}$ \( 1 + 10 T + 128 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_ey
59$D_{4}$ \( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.59.au_ih
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.67.a_es
71$D_{4}$ \( 1 - 4 T + 143 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_fn
73$D_{4}$ \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_cu
79$D_{4}$ \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_le
83$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fs
89$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \) 2.89.a_fv
97$D_{4}$ \( 1 + 2 T + 120 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_eq
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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

−11.81280704555777175007360061082, −11.17497865893224840085333193319, −10.70111419310084895450262454155, −10.01402088090004713465850639378, −9.764901978379504904420787430943, −9.329692548641521249568089352375, −8.894202601271345561628108399104, −8.118839625677205964674179257176, −7.52722054536155113207679277687, −6.98720415408259830572734549482, −6.69649917282905943071590583602, −6.44725846206581020854653092934, −5.58837178854773700140581689069, −5.20184647625128074553158387134, −4.36172788927928137817911118524, −4.00117127050492294209886581282, −3.39185724954111398066867151906, −3.38240342252298975101634836917, −2.22456788844258673958989510895, −0.923154092891884846354495886643, 0.923154092891884846354495886643, 2.22456788844258673958989510895, 3.38240342252298975101634836917, 3.39185724954111398066867151906, 4.00117127050492294209886581282, 4.36172788927928137817911118524, 5.20184647625128074553158387134, 5.58837178854773700140581689069, 6.44725846206581020854653092934, 6.69649917282905943071590583602, 6.98720415408259830572734549482, 7.52722054536155113207679277687, 8.118839625677205964674179257176, 8.894202601271345561628108399104, 9.329692548641521249568089352375, 9.764901978379504904420787430943, 10.01402088090004713465850639378, 10.70111419310084895450262454155, 11.17497865893224840085333193319, 11.81280704555777175007360061082

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