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Properties

Label 2-379050-1.1-c1-0-100
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 − 5·11-s − 12-s + 3·13-s − 14-s + 16-s + 5·17-s − 18-s − 21-s + 5·22-s − 7·23-s + 24-s − 3·26-s − 27-s + 28-s + 3·29-s − 6·31-s − 32-s + 5·33-s − 5·34-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 − 1.50·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.218·21-s + 1.06·22-s − 1.45·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s + 0.188·28-s + 0.557·29-s − 1.07·31-s − 0.176·32-s + 0.870·33-s − 0.857·34-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−12.40678579507880, −12.35486903854615, −11.68418525598597, −11.22477967812837, −10.92124294316204, −10.38469022763463, −9.926735697933253, −9.897904835231918, −9.026053789524174, −8.504307770633018, −8.000344018625627, −7.936027067857637, −7.232360634240603, −6.853964714550390, −6.093325494306304, −5.832685344347588, −5.304778443049043, −4.960160795250128, −4.200712391247041, −3.638667864308603, −3.101880570704624, −2.508413634004581, −1.751280494698784, −1.458121905171455, −0.5952136470083808, 0, 0.5952136470083808, 1.458121905171455, 1.751280494698784, 2.508413634004581, 3.101880570704624, 3.638667864308603, 4.200712391247041, 4.960160795250128, 5.304778443049043, 5.832685344347588, 6.093325494306304, 6.853964714550390, 7.232360634240603, 7.936027067857637, 8.000344018625627, 8.504307770633018, 9.026053789524174, 9.897904835231918, 9.926735697933253, 10.38469022763463, 10.92124294316204, 11.22477967812837, 11.68418525598597, 12.35486903854615, 12.40678579507880

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