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Properties

Label 2-379050-1.1-c1-0-147
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 + 4·11-s − 12-s + 14-s + 16-s + 4·17-s − 18-s + 21-s − 4·22-s − 6·23-s + 24-s − 27-s − 28-s + 10·29-s − 10·31-s − 32-s − 4·33-s − 4·34-s + 36-s − 2·37-s + 2·41-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 + 1.20·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.218·21-s − 0.852·22-s − 1.25·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s + 1.85·29-s − 1.79·31-s − 0.176·32-s − 0.696·33-s − 0.685·34-s + 1/6·36-s − 0.328·37-s + 0.312·41-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.58494871495302, −12.22585544563288, −11.72452478609101, −11.35538987780857, −10.93763408851220, −10.28382900373110, −10.04742394153965, −9.561657169699860, −9.191893263624255, −8.682580044383822, −8.045044570066738, −7.823588480779137, −7.099461285153223, −6.729867795189807, −6.281776216361311, −5.940556665557528, −5.332969450253987, −4.809909590029855, −4.156925659012163, −3.573974647193079, −3.307638458764154, −2.422926285970708, −1.842911843329970, −1.278128928736069, −0.7313399777356255, 0, 0.7313399777356255, 1.278128928736069, 1.842911843329970, 2.422926285970708, 3.307638458764154, 3.573974647193079, 4.156925659012163, 4.809909590029855, 5.332969450253987, 5.940556665557528, 6.281776216361311, 6.729867795189807, 7.099461285153223, 7.823588480779137, 8.045044570066738, 8.682580044383822, 9.191893263624255, 9.561657169699860, 10.04742394153965, 10.28382900373110, 10.93763408851220, 11.35538987780857, 11.72452478609101, 12.22585544563288, 12.58494871495302

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