Properties

Label 2-139650-1.1-c1-0-10
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 − 2·11-s − 12-s − 4·13-s + 16-s − 3·17-s − 18-s − 19-s + 2·22-s − 3·23-s + 24-s + 4·26-s − 27-s − 6·29-s + 3·31-s − 32-s + 2·33-s + 3·34-s + 36-s + 8·37-s + 38-s + 4·39-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 − 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s + 0.426·22-s − 0.625·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 1.11·29-s + 0.538·31-s − 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s + 1.31·37-s + 0.162·38-s + 0.640·39-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: $\chi_{139650} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5899431158\)
\(L(\frac12)\) \(\approx\) \(0.5899431158\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.20576376547836, −12.77064544221089, −12.52812270739446, −11.87382056044624, −11.28943901050240, −11.09677553927128, −10.54662194234655, −9.984331661372618, −9.535461397667327, −9.283633477194050, −8.516220218915895, −7.823355459157492, −7.750074379581554, −7.045713884642333, −6.543208328781398, −6.050267426521497, −5.439618508066291, −5.039725956688195, −4.193286401235639, −3.995773837134452, −2.814071606092355, −2.479473548611151, −1.915889208808026, −1.007238015287822, −0.3034177428105993, 0.3034177428105993, 1.007238015287822, 1.915889208808026, 2.479473548611151, 2.814071606092355, 3.995773837134452, 4.193286401235639, 5.039725956688195, 5.439618508066291, 6.050267426521497, 6.543208328781398, 7.045713884642333, 7.750074379581554, 7.823355459157492, 8.516220218915895, 9.283633477194050, 9.535461397667327, 9.984331661372618, 10.54662194234655, 11.09677553927128, 11.28943901050240, 11.87382056044624, 12.52812270739446, 12.77064544221089, 13.20576376547836

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