Properties

Label 4-1710e2-1.1-c1e2-0-12
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 5-s + 6·7-s − 8-s − 10-s − 2·11-s − 2·13-s + 6·14-s − 16-s + 2·17-s + 19-s − 2·22-s + 23-s − 2·26-s + 8·31-s + 2·34-s − 6·35-s + 18·37-s + 38-s + 40-s + 41-s + 10·43-s + 46-s + 13·49-s + 3·53-s + 2·55-s − 6·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s + 2.26·7-s − 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s + 1.60·14-s − 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.426·22-s + 0.208·23-s − 0.392·26-s + 1.43·31-s + 0.342·34-s − 1.01·35-s + 2.95·37-s + 0.162·38-s + 0.158·40-s + 0.156·41-s + 1.52·43-s + 0.147·46-s + 13/7·49-s + 0.412·53-s + 0.269·55-s − 0.801·56-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

\(L(1)\) \(\approx\) \(4.283517511\)
\(L(\frac12)\) \(\approx\) \(4.283517511\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 5 T - 64 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−9.389156539245316422905992646167, −9.280851674372576138784793775995, −8.349408563389744036958359762603, −8.301522288546477268076153838221, −8.029103554375264075708749801953, −7.65268048739800356716683209815, −7.25361188835612940501957004921, −6.79153371755695353898350371960, −6.19152075228147021044937933238, −5.61959985657414192425337522229, −5.42809001240074574679078417707, −4.94429593282086487103943385517, −4.58309503670190111370410861752, −4.15940121884563160129347448406, −3.99593126082916570078833286023, −2.99579366212051413716056610048, −2.60840458665968772588756664149, −2.20861908644391268531740518711, −1.28234868870439648036751210380, −0.791379575076051957947039970828, 0.791379575076051957947039970828, 1.28234868870439648036751210380, 2.20861908644391268531740518711, 2.60840458665968772588756664149, 2.99579366212051413716056610048, 3.99593126082916570078833286023, 4.15940121884563160129347448406, 4.58309503670190111370410861752, 4.94429593282086487103943385517, 5.42809001240074574679078417707, 5.61959985657414192425337522229, 6.19152075228147021044937933238, 6.79153371755695353898350371960, 7.25361188835612940501957004921, 7.65268048739800356716683209815, 8.029103554375264075708749801953, 8.301522288546477268076153838221, 8.349408563389744036958359762603, 9.280851674372576138784793775995, 9.389156539245316422905992646167

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