Properties

Label 4-930e2-1.1-c1e2-0-8
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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  + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 7-s + 4·8-s − 2·10-s − 5·11-s − 3·12-s − 4·13-s + 2·14-s + 15-s + 5·16-s + 2·17-s + 2·19-s − 3·20-s − 21-s − 10·22-s − 8·23-s − 4·24-s − 8·26-s + 27-s + 3·28-s + 6·29-s + 2·30-s + 7·31-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1.41·8-s − 0.632·10-s − 1.50·11-s − 0.866·12-s − 1.10·13-s + 0.534·14-s + 0.258·15-s + 5/4·16-s + 0.485·17-s + 0.458·19-s − 0.670·20-s − 0.218·21-s − 2.13·22-s − 1.66·23-s − 0.816·24-s − 1.56·26-s + 0.192·27-s + 0.566·28-s + 1.11·29-s + 0.365·30-s + 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.021650497\)
\(L(\frac12)\) \(\approx\) \(3.021650497\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
31$C_2$ \( 1 - 7 T + p T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−10.27435505761476451009128576905, −10.26766778397209428236956838925, −9.679312627270711473262758799626, −9.040606803200149543593425769758, −8.223942431847154701535969405585, −8.107670220484163438870640559764, −7.55644136112322467177848497916, −7.40414112668966146621646301903, −6.77213598447190992149365961721, −6.12297997229029298466998079656, −5.94985397647227030359309795216, −5.37593847345605006537868557699, −4.91328561041160248966696011618, −4.70482354311543036645031617494, −4.15903527634336895777912034697, −3.58583723919905213035612411632, −2.70545766964029665005559027530, −2.68985758998949375222145606903, −1.83497885094960504484096930921, −0.65681548417145443507407451125, 0.65681548417145443507407451125, 1.83497885094960504484096930921, 2.68985758998949375222145606903, 2.70545766964029665005559027530, 3.58583723919905213035612411632, 4.15903527634336895777912034697, 4.70482354311543036645031617494, 4.91328561041160248966696011618, 5.37593847345605006537868557699, 5.94985397647227030359309795216, 6.12297997229029298466998079656, 6.77213598447190992149365961721, 7.40414112668966146621646301903, 7.55644136112322467177848497916, 8.107670220484163438870640559764, 8.223942431847154701535969405585, 9.040606803200149543593425769758, 9.679312627270711473262758799626, 10.26766778397209428236956838925, 10.27435505761476451009128576905

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