Properties

Label 4-1183e2-1.1-c1e2-0-9
Degree $4$
Conductor $1399489$
Sign $1$
Analytic cond. $89.2326$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 2·7-s − 6·11-s − 2·12-s − 3·16-s − 12·17-s − 4·19-s − 4·21-s + 6·23-s − 7·25-s − 2·27-s + 2·28-s − 6·29-s + 2·31-s − 12·33-s + 14·37-s + 10·43-s + 6·44-s − 12·47-s − 6·48-s + 3·49-s − 24·51-s − 6·53-s − 8·57-s − 18·59-s − 20·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 0.755·7-s − 1.80·11-s − 0.577·12-s − 3/4·16-s − 2.91·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s − 7/5·25-s − 0.384·27-s + 0.377·28-s − 1.11·29-s + 0.359·31-s − 2.08·33-s + 2.30·37-s + 1.52·43-s + 0.904·44-s − 1.75·47-s − 0.866·48-s + 3/7·49-s − 3.36·51-s − 0.824·53-s − 1.05·57-s − 2.34·59-s − 2.56·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1399489\)    =    \(7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(89.2326\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1399489,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bc
17$D_{4}$ \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cp
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_bk
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.41.a_cd
43$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dg
47$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_de
53$D_{4}$ \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_cp
59$D_{4}$ \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_ho
61$D_{4}$ \( 1 + 20 T + 195 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.61.u_hn
67$D_{4}$ \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_ee
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$D_{4}$ \( 1 + 4 T + 123 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_et
79$D_{4}$ \( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.79.aw_js
83$D_{4}$ \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_fs
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gk
97$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_dy
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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

−9.360438878349297248235224639730, −9.318431514677108533108845399192, −8.649628853872145341367351689353, −8.554796684694620869248593087602, −7.983608496186141154462739630124, −7.51821351177049365235861067884, −7.38059187606478749895460845120, −6.52901228257874317659307463150, −6.18826341062502788637453866112, −6.01847552641775700461429554518, −5.08770078656462847327918556255, −4.68550777932376371925645836081, −4.36832608437226875511155361649, −3.91086875357420949604621609186, −2.99324455808367928275809900809, −2.79950106455141729889317959166, −2.37190935564413613059173871538, −1.82481648344398235946456611792, 0, 0, 1.82481648344398235946456611792, 2.37190935564413613059173871538, 2.79950106455141729889317959166, 2.99324455808367928275809900809, 3.91086875357420949604621609186, 4.36832608437226875511155361649, 4.68550777932376371925645836081, 5.08770078656462847327918556255, 6.01847552641775700461429554518, 6.18826341062502788637453866112, 6.52901228257874317659307463150, 7.38059187606478749895460845120, 7.51821351177049365235861067884, 7.983608496186141154462739630124, 8.554796684694620869248593087602, 8.649628853872145341367351689353, 9.318431514677108533108845399192, 9.360438878349297248235224639730

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