Properties

Label 2-327990-1.1-c1-0-6
Degree $2$
Conductor $327990$
Sign $1$
Analytic cond. $2619.01$
Root an. cond. $51.1762$
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 − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 3·11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 2·17-s − 18-s − 20-s + 21-s + 3·22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s + 0.639·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(327990\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(2619.01\)
Root analytic conductor: \(51.1762\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{327990} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8465001372\)
\(L(\frac12)\) \(\approx\) \(0.8465001372\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
29 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−12.50238591815099, −11.96205478836339, −11.72462017724444, −11.18833604952037, −10.79949903997415, −10.16519119425072, −10.05535697959478, −9.521347235020503, −8.939573729376565, −8.345586074005899, −8.059800357496712, −7.608703506535340, −7.011225077063443, −6.563676790259057, −6.286763474027908, −5.515352537536455, −5.165911360421730, −4.535692163532943, −4.084427825944641, −3.383817206378997, −2.746330616354426, −2.391796849394598, −1.633548006741377, −0.8145271653007592, −0.3752109432305095, 0.3752109432305095, 0.8145271653007592, 1.633548006741377, 2.391796849394598, 2.746330616354426, 3.383817206378997, 4.084427825944641, 4.535692163532943, 5.165911360421730, 5.515352537536455, 6.286763474027908, 6.563676790259057, 7.011225077063443, 7.608703506535340, 8.059800357496712, 8.345586074005899, 8.939573729376565, 9.521347235020503, 10.05535697959478, 10.16519119425072, 10.79949903997415, 11.18833604952037, 11.72462017724444, 11.96205478836339, 12.50238591815099

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