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Properties

Label 4-3675e2-1.1-c1e2-0-3
Degree $4$
Conductor $13505625$
Sign $1$
Analytic cond. $861.130$
Root an. cond. $5.41710$
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  − 3·2-s + 2·3-s + 4·4-s − 6·6-s − 3·8-s + 3·9-s − 2·11-s + 8·12-s + 6·13-s + 3·16-s − 2·17-s − 9·18-s + 2·19-s + 6·22-s − 10·23-s − 6·24-s − 18·26-s + 4·27-s − 4·29-s − 6·32-s − 4·33-s + 6·34-s + 12·36-s − 2·37-s − 6·38-s + 12·39-s + 16·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 2·4-s − 2.44·6-s − 1.06·8-s + 9-s − 0.603·11-s + 2.30·12-s + 1.66·13-s + 3/4·16-s − 0.485·17-s − 2.12·18-s + 0.458·19-s + 1.27·22-s − 2.08·23-s − 1.22·24-s − 3.53·26-s + 0.769·27-s − 0.742·29-s − 1.06·32-s − 0.696·33-s + 1.02·34-s + 2·36-s − 0.328·37-s − 0.973·38-s + 1.92·39-s + 2.49·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(861.130\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237796079\)
\(L(\frac12)\) \(\approx\) \(1.237796079\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
11$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_d
13$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_be
17$D_{4}$ \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_ak
19$D_{4}$ \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bi
23$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.23.k_ct
29$D_{4}$ \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_r
31$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.31.a_bq
37$D_{4}$ \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cd
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$D_{4}$ \( 1 + 12 T + 117 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_en
47$D_{4}$ \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_ek
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_es
61$C_4$ \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_acc
67$D_{4}$ \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_eb
71$D_{4}$ \( 1 + 10 T + 147 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_fr
73$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fm
79$D_{4}$ \( 1 - 8 T + 169 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_gn
83$D_{4}$ \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_ic
89$D_{4}$ \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_fe
97$D_{4}$ \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_je
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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

−8.668055328510231159726624731388, −8.563362030179154189249851116051, −8.005408716677935806331482791598, −7.918994549516426320785882137870, −7.37280407324740512629002937130, −7.36442530273358895091899432444, −6.74483141747641033482288038539, −6.11830722046096385990745403622, −5.84504124464702679918995066724, −5.72112016290433679747410834397, −4.79122456603031799517233706677, −4.43070634832946540365962644354, −3.88713232929704353412486717113, −3.48663380880317772519994741935, −3.25210509714808960555714928455, −2.42488400560722441349749799093, −2.03701692734220149177413444035, −1.75182512055352581626648230639, −0.912485903142973144279626117373, −0.56063986540561002413308348445, 0.56063986540561002413308348445, 0.912485903142973144279626117373, 1.75182512055352581626648230639, 2.03701692734220149177413444035, 2.42488400560722441349749799093, 3.25210509714808960555714928455, 3.48663380880317772519994741935, 3.88713232929704353412486717113, 4.43070634832946540365962644354, 4.79122456603031799517233706677, 5.72112016290433679747410834397, 5.84504124464702679918995066724, 6.11830722046096385990745403622, 6.74483141747641033482288038539, 7.36442530273358895091899432444, 7.37280407324740512629002937130, 7.918994549516426320785882137870, 8.005408716677935806331482791598, 8.563362030179154189249851116051, 8.668055328510231159726624731388

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