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Properties

Label 2-379050-1.1-c1-0-4
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 $0$

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 − 6·11-s − 12-s − 6·13-s − 14-s + 16-s + 3·17-s − 18-s − 21-s + 6·22-s − 6·23-s + 24-s + 6·26-s − 27-s + 28-s + 4·29-s + 4·31-s − 32-s + 6·33-s − 3·34-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 − 1.80·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.218·21-s + 1.27·22-s − 1.25·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.742·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s − 0.514·34-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: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3786404901\)
\(L(\frac12)\) \(\approx\) \(0.3786404901\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.33166224215666, −11.90046321039483, −11.75133276337797, −11.02132920011168, −10.43329679857611, −10.18585798593637, −10.03290977951440, −9.520086014883924, −8.675580777952609, −8.366561503372869, −7.800424866256439, −7.633905085868495, −6.885686263608748, −6.814991748357541, −5.741796182528328, −5.586070495235243, −5.113689372432309, −4.632971816294032, −4.057051972395298, −3.258073049698218, −2.618447532332879, −2.333007851389825, −1.711933978577850, −0.8907501180097839, −0.2188860171412741, 0.2188860171412741, 0.8907501180097839, 1.711933978577850, 2.333007851389825, 2.618447532332879, 3.258073049698218, 4.057051972395298, 4.632971816294032, 5.113689372432309, 5.586070495235243, 5.741796182528328, 6.814991748357541, 6.885686263608748, 7.633905085868495, 7.800424866256439, 8.366561503372869, 8.675580777952609, 9.520086014883924, 10.03290977951440, 10.18585798593637, 10.43329679857611, 11.02132920011168, 11.75133276337797, 11.90046321039483, 12.33166224215666

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