Properties

Label 4-1710e2-1.1-c1e2-0-1
Degree $4$
Conductor $2924100$
Sign $1$
Analytic cond. $186.443$
Root an. cond. $3.69518$
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  − 2-s + 5-s − 10·7-s + 8-s − 10-s − 2·11-s − 6·13-s + 10·14-s − 16-s + 4·17-s − 19-s + 2·22-s + 7·23-s + 6·26-s + 6·29-s − 4·34-s − 10·35-s + 14·37-s + 38-s + 40-s − 5·41-s − 6·43-s − 7·46-s − 8·47-s + 61·49-s + 11·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s − 3.77·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 1.66·13-s + 2.67·14-s − 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.426·22-s + 1.45·23-s + 1.17·26-s + 1.11·29-s − 0.685·34-s − 1.69·35-s + 2.30·37-s + 0.162·38-s + 0.158·40-s − 0.780·41-s − 0.914·43-s − 1.03·46-s − 1.16·47-s + 61/7·49-s + 1.51·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2924100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(186.443\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1710} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2924100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4658129063\)
\(L(\frac12)\) \(\approx\) \(0.4658129063\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−9.662880775712459049022105377144, −9.266738264866538870634634773207, −8.844054411717622077399636632895, −8.674753056178696405250714619088, −7.79775482413170400776768667274, −7.43366043197413088061239649776, −7.26672082479702954143164479571, −6.67138538879137382532202686114, −6.30450335167263285415955961060, −6.21472855405825764037049207558, −5.62887984335819565507149161373, −4.92603064421680306845631958872, −4.81316911907771002250255011138, −3.85939317970140680894921823197, −3.44533408316776937831906264339, −2.89382931247378853677318609190, −2.79838686637030373939818071606, −2.22309201653286083952235233638, −0.921348307351019644856337763920, −0.36994564203229181857154935077, 0.36994564203229181857154935077, 0.921348307351019644856337763920, 2.22309201653286083952235233638, 2.79838686637030373939818071606, 2.89382931247378853677318609190, 3.44533408316776937831906264339, 3.85939317970140680894921823197, 4.81316911907771002250255011138, 4.92603064421680306845631958872, 5.62887984335819565507149161373, 6.21472855405825764037049207558, 6.30450335167263285415955961060, 6.67138538879137382532202686114, 7.26672082479702954143164479571, 7.43366043197413088061239649776, 7.79775482413170400776768667274, 8.674753056178696405250714619088, 8.844054411717622077399636632895, 9.266738264866538870634634773207, 9.662880775712459049022105377144

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