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Properties

Label 2-379050-1.1-c1-0-80
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 − 14-s + 16-s − 6·17-s − 18-s − 21-s + 5·22-s − 5·23-s + 24-s − 27-s + 28-s + 3·29-s + 4·31-s − 32-s + 5·33-s + 6·34-s + 36-s + 6·37-s + 3·41-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 − 1.50·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.218·21-s + 1.06·22-s − 1.04·23-s + 0.204·24-s − 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.718·31-s − 0.176·32-s + 0.870·33-s + 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.468·41-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
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

−12.78116742748844, −11.97618279087532, −11.78349225700378, −11.25037163930903, −10.91275359582987, −10.38170423473093, −10.07829838221083, −9.727418958237302, −9.009514626957553, −8.554096989134119, −8.132802552725191, −7.773572461577458, −7.302855686131919, −6.643783422636726, −6.341992014987825, −5.850824980761383, −5.196851453702131, −4.819550508542991, −4.377095792131755, −3.708928052436989, −2.951040055375272, −2.404038566003100, −2.048422111494846, −1.326011441016407, −0.5415984467613382, 0, 0.5415984467613382, 1.326011441016407, 2.048422111494846, 2.404038566003100, 2.951040055375272, 3.708928052436989, 4.377095792131755, 4.819550508542991, 5.196851453702131, 5.850824980761383, 6.341992014987825, 6.643783422636726, 7.302855686131919, 7.773572461577458, 8.132802552725191, 8.554096989134119, 9.009514626957553, 9.727418958237302, 10.07829838221083, 10.38170423473093, 10.91275359582987, 11.25037163930903, 11.78349225700378, 11.97618279087532, 12.78116742748844

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