Properties

Label 4-1617e2-1.1-c1e2-0-7
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  − 2·2-s + 2·3-s + 4-s − 4·5-s − 4·6-s + 3·9-s + 8·10-s + 2·11-s + 2·12-s + 4·13-s − 8·15-s + 16-s − 6·17-s − 6·18-s + 6·19-s − 4·20-s − 4·22-s − 14·23-s + 2·25-s − 8·26-s + 4·27-s − 2·29-s + 16·30-s + 2·32-s + 4·33-s + 12·34-s + 3·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.78·5-s − 1.63·6-s + 9-s + 2.52·10-s + 0.603·11-s + 0.577·12-s + 1.10·13-s − 2.06·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s + 1.37·19-s − 0.894·20-s − 0.852·22-s − 2.91·23-s + 2/5·25-s − 1.56·26-s + 0.769·27-s − 0.371·29-s + 2.92·30-s + 0.353·32-s + 0.696·33-s + 2.05·34-s + 1/2·36-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$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 14 T + 183 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−8.925244273161343507784713384625, −8.920776835107114659191518387096, −8.274631162935441020542379527799, −8.205627343334746979130113953587, −7.73432823471529795765853539423, −7.66628361659559764926271557384, −7.08900222400342542256647763104, −6.50291801372496204300094668652, −6.18925706956845655307866109056, −5.69652527313945590097875835864, −4.69304038187189757170150256612, −4.50364216333560103291685329939, −3.86750561940473517655195679508, −3.63038134202443877586541385678, −3.33317843839701959790536576400, −2.55269324795494843239122022897, −1.69210111069279353343833228513, −1.41942960474794095224470410930, 0, 0, 1.41942960474794095224470410930, 1.69210111069279353343833228513, 2.55269324795494843239122022897, 3.33317843839701959790536576400, 3.63038134202443877586541385678, 3.86750561940473517655195679508, 4.50364216333560103291685329939, 4.69304038187189757170150256612, 5.69652527313945590097875835864, 6.18925706956845655307866109056, 6.50291801372496204300094668652, 7.08900222400342542256647763104, 7.66628361659559764926271557384, 7.73432823471529795765853539423, 8.205627343334746979130113953587, 8.274631162935441020542379527799, 8.920776835107114659191518387096, 8.925244273161343507784713384625

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