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Properties

Label 2-139650-1.1-c1-0-122
Degree $2$
Conductor $139650$
Sign $1$
Analytic cond. $1115.11$
Root an. cond. $33.3932$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s − 2·13-s + 16-s − 2·17-s + 18-s − 19-s + 4·22-s + 8·23-s + 24-s − 2·26-s + 27-s + 6·29-s + 32-s + 4·33-s − 2·34-s + 36-s + 10·37-s − 38-s − 2·39-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.852·22-s + 1.66·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s − 0.162·38-s − 0.320·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(139650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1115.11\)
Root analytic conductor: \(33.3932\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.429430615\)
\(L(\frac12)\) \(\approx\) \(7.429430615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
19 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.47268089798205, −12.96004462687820, −12.54039875697666, −12.09270818582494, −11.56423041762642, −11.07522705680170, −10.67523548250606, −10.02049399238706, −9.411246816044195, −9.123967251126966, −8.618319490810136, −7.975046922702206, −7.414383632900852, −7.000433709350935, −6.413470695499493, −6.123983828333315, −5.292374551518584, −4.705899293183568, −4.327205066020997, −3.838634368940429, −3.076973591464313, −2.693478504101489, −2.100550350650887, −1.266140066212923, −0.7409049797617979, 0.7409049797617979, 1.266140066212923, 2.100550350650887, 2.693478504101489, 3.076973591464313, 3.838634368940429, 4.327205066020997, 4.705899293183568, 5.292374551518584, 6.123983828333315, 6.413470695499493, 7.000433709350935, 7.414383632900852, 7.975046922702206, 8.618319490810136, 9.123967251126966, 9.411246816044195, 10.02049399238706, 10.67523548250606, 11.07522705680170, 11.56423041762642, 12.09270818582494, 12.54039875697666, 12.96004462687820, 13.47268089798205

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