Properties

Label 4-1617e2-1.1-c1e2-0-9
Degree $4$
Conductor $2614689$
Sign $1$
Analytic cond. $166.714$
Root an. cond. $3.59330$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

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 + 2·13-s + 3·16-s − 12·17-s − 9·18-s + 4·19-s + 6·22-s + 4·23-s − 6·24-s − 5·25-s − 6·26-s + 4·27-s − 16·29-s − 8·31-s − 6·32-s − 4·33-s + 36·34-s + 12·36-s − 10·37-s − 12·38-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 + 0.554·13-s + 3/4·16-s − 2.91·17-s − 2.12·18-s + 0.917·19-s + 1.27·22-s + 0.834·23-s − 1.22·24-s − 25-s − 1.17·26-s + 0.769·27-s − 2.97·29-s − 1.43·31-s − 1.06·32-s − 0.696·33-s + 6.17·34-s + 2·36-s − 1.64·37-s − 1.94·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2614689\)    =    \(3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(166.714\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2614689,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 16 T + 117 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.136129125392656469050322013341, −8.807030341900377071986106821698, −8.458543213076420806943974649736, −8.410384676094100896686091956837, −7.60782286100225971478383860507, −7.52275297217827678920123400387, −7.03252398659051344961708471276, −6.84612894136881739739190128481, −6.08458139083664908339960167012, −5.62692514355561985503532827542, −5.08833377946787577344317188999, −4.54919738362808015152712971308, −3.92670638612145572212418942778, −3.40230264114730994288873085529, −3.12806557848555480053285195221, −2.16099573002650604749150685811, −1.84069596447255548673900203893, −1.51088916595908606697405159221, 0, 0, 1.51088916595908606697405159221, 1.84069596447255548673900203893, 2.16099573002650604749150685811, 3.12806557848555480053285195221, 3.40230264114730994288873085529, 3.92670638612145572212418942778, 4.54919738362808015152712971308, 5.08833377946787577344317188999, 5.62692514355561985503532827542, 6.08458139083664908339960167012, 6.84612894136881739739190128481, 7.03252398659051344961708471276, 7.52275297217827678920123400387, 7.60782286100225971478383860507, 8.410384676094100896686091956837, 8.458543213076420806943974649736, 8.807030341900377071986106821698, 9.136129125392656469050322013341

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