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Properties

Label 2-139650-1.1-c1-0-109
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 $1$

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 + 5·13-s + 16-s + 3·17-s − 18-s − 19-s + 4·22-s + 6·23-s + 24-s − 5·26-s − 27-s − 9·29-s − 31-s − 32-s + 4·33-s − 3·34-s + 36-s − 7·37-s + 38-s − 5·39-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 − 1.20·11-s − 0.288·12-s + 1.38·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s + 0.852·22-s + 1.25·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 1.67·29-s − 0.179·31-s − 0.176·32-s + 0.696·33-s − 0.514·34-s + 1/6·36-s − 1.15·37-s + 0.162·38-s − 0.800·39-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−13.29997818186028, −13.09461089004871, −12.90890317928179, −12.04591760956466, −11.60272079345008, −11.05563040282565, −10.92299751564277, −10.17311761765265, −10.03627365640714, −9.273082506788486, −8.735755827864190, −8.394178246179453, −7.749682737022367, −7.365587238239891, −6.792327716301993, −6.286461692798310, −5.638462574441178, −5.340630310739022, −4.784953092425213, −3.919810089495123, −3.328171757531927, −2.939747807894264, −1.902890726883285, −1.545958026734584, −0.7159312581604934, 0, 0.7159312581604934, 1.545958026734584, 1.902890726883285, 2.939747807894264, 3.328171757531927, 3.919810089495123, 4.784953092425213, 5.340630310739022, 5.638462574441178, 6.286461692798310, 6.792327716301993, 7.365587238239891, 7.749682737022367, 8.394178246179453, 8.735755827864190, 9.273082506788486, 10.03627365640714, 10.17311761765265, 10.92299751564277, 11.05563040282565, 11.60272079345008, 12.04591760956466, 12.90890317928179, 13.09461089004871, 13.29997818186028

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