Properties

Label 4-546e2-1.1-c1e2-0-3
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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 − 3·3-s + 3·6-s + 7-s + 8-s + 6·9-s + 3·11-s − 7·13-s − 14-s − 16-s − 3·17-s − 6·18-s + 7·19-s − 3·21-s − 3·22-s − 12·23-s − 3·24-s − 2·25-s + 7·26-s − 9·27-s + 3·29-s + 8·31-s − 9·33-s + 3·34-s − 12·37-s − 7·38-s + 21·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.904·11-s − 1.94·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 1.41·18-s + 1.60·19-s − 0.654·21-s − 0.639·22-s − 2.50·23-s − 0.612·24-s − 2/5·25-s + 1.37·26-s − 1.73·27-s + 0.557·29-s + 1.43·31-s − 1.56·33-s + 0.514·34-s − 1.97·37-s − 1.13·38-s + 3.36·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3540554351\)
\(L(\frac12)\) \(\approx\) \(0.3540554351\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_2$ \( 1 + 7 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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

−11.18768309659210800717838757323, −10.26446171288301205093854652485, −10.17552540039730066391807440924, −9.872861492526706940933631951139, −9.591764581077383762854464237297, −8.801074340813380316353018812772, −8.401915404565964842425635768301, −7.80493686535132373946583636313, −7.37750264933425711881560841239, −6.83558620697021117082112387211, −6.70584308427017024290162313096, −5.82342839953051674251294832005, −5.60125147789500580165917733627, −4.95975274207266589749162577806, −4.48526993614419494642454052670, −4.18281535533042208124895976524, −3.21746659207947276617197482878, −2.13712628117161676740446058654, −1.53150468806062065916599278994, −0.44560280148088547958625287159, 0.44560280148088547958625287159, 1.53150468806062065916599278994, 2.13712628117161676740446058654, 3.21746659207947276617197482878, 4.18281535533042208124895976524, 4.48526993614419494642454052670, 4.95975274207266589749162577806, 5.60125147789500580165917733627, 5.82342839953051674251294832005, 6.70584308427017024290162313096, 6.83558620697021117082112387211, 7.37750264933425711881560841239, 7.80493686535132373946583636313, 8.401915404565964842425635768301, 8.801074340813380316353018812772, 9.591764581077383762854464237297, 9.872861492526706940933631951139, 10.17552540039730066391807440924, 10.26446171288301205093854652485, 11.18768309659210800717838757323

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