Properties

Label 2-139650-1.1-c1-0-108
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 − 2·11-s − 12-s − 5·13-s + 16-s − 4·17-s + 18-s + 19-s − 2·22-s − 4·23-s − 24-s − 5·26-s − 27-s − 2·29-s − 4·31-s + 32-s + 2·33-s − 4·34-s + 36-s + 38-s + 5·39-s + 9·41-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 − 0.603·11-s − 0.288·12-s − 1.38·13-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.229·19-s − 0.426·22-s − 0.834·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.348·33-s − 0.685·34-s + 1/6·36-s + 0.162·38-s + 0.800·39-s + 1.40·41-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: $\chi_{139650} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.44004516767575, −13.17869385768086, −12.73213933446700, −12.23406491237329, −11.77558665201911, −11.37704772945735, −10.91132560874778, −10.23255907991971, −10.07608102677841, −9.347091288097090, −8.910463074278537, −8.042357070204597, −7.711234678188953, −7.138961284199136, −6.741889593238708, −6.127249328113266, −5.592213149923550, −5.134848815736173, −4.688477029547065, −4.159695480953093, −3.571370763641725, −2.821003822633459, −2.202652175683786, −1.858383086442463, −0.7155357015459965, 0, 0.7155357015459965, 1.858383086442463, 2.202652175683786, 2.821003822633459, 3.571370763641725, 4.159695480953093, 4.688477029547065, 5.134848815736173, 5.592213149923550, 6.127249328113266, 6.741889593238708, 7.138961284199136, 7.711234678188953, 8.042357070204597, 8.910463074278537, 9.347091288097090, 10.07608102677841, 10.23255907991971, 10.91132560874778, 11.37704772945735, 11.77558665201911, 12.23406491237329, 12.73213933446700, 13.17869385768086, 13.44004516767575

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