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Properties

Label 2-379050-1.1-c1-0-132
Degree $2$
Conductor $379050$
Sign $-1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
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 − 7-s − 8-s + 9-s + 2·11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 21-s − 2·22-s + 7·23-s + 24-s + 26-s − 27-s − 28-s − 29-s − 3·31-s − 32-s − 2·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.218·21-s − 0.426·22-s + 1.45·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.185·29-s − 0.538·31-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.78871957320027, −12.20839543759235, −11.48610230425106, −11.36347124685343, −11.07137166769778, −10.33980864679877, −10.00090838834415, −9.537043381169464, −9.145458885445631, −8.737546149555730, −8.169409732035112, −7.615314717999213, −7.153344721948751, −6.795448545386035, −6.254665881337980, −5.969203280288320, −5.219575870081797, −4.837565831791209, −4.274633121836703, −3.589624945539954, −3.130282965786974, −2.534616994275431, −1.841932664611132, −1.282038324854902, −0.6844024677048519, 0, 0.6844024677048519, 1.282038324854902, 1.841932664611132, 2.534616994275431, 3.130282965786974, 3.589624945539954, 4.274633121836703, 4.837565831791209, 5.219575870081797, 5.969203280288320, 6.254665881337980, 6.795448545386035, 7.153344721948751, 7.615314717999213, 8.169409732035112, 8.737546149555730, 9.145458885445631, 9.537043381169464, 10.00090838834415, 10.33980864679877, 11.07137166769778, 11.36347124685343, 11.48610230425106, 12.20839543759235, 12.78871957320027

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