Properties

Label 2-327990-1.1-c1-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 3·14-s − 15-s + 16-s − 2·17-s + 18-s + 20-s − 3·21-s + 22-s + 5·23-s − 24-s + 25-s − 26-s − 27-s + 3·28-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 + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.04·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.566·28-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: \(1\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 + T \)
29 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
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

−12.95252978026018, −12.17139959950003, −11.97162514242872, −11.60993067796149, −11.04772632033350, −10.62267081861700, −10.40661595965745, −9.695200556370700, −9.183000202610969, −8.692819394033863, −8.217634304401881, −7.608171718258969, −7.160630907327632, −6.692920984458867, −6.283854020273718, −5.677630867825213, −5.172991915727652, −4.925533159814148, −4.407943741082727, −3.917223644367042, −3.217676838791340, −2.599939931596443, −2.045506988090933, −1.446638982254651, −1.012638619370529, 0, 1.012638619370529, 1.446638982254651, 2.045506988090933, 2.599939931596443, 3.217676838791340, 3.917223644367042, 4.407943741082727, 4.925533159814148, 5.172991915727652, 5.677630867825213, 6.283854020273718, 6.692920984458867, 7.160630907327632, 7.608171718258969, 8.217634304401881, 8.692819394033863, 9.183000202610969, 9.695200556370700, 10.40661595965745, 10.62267081861700, 11.04772632033350, 11.60993067796149, 11.97162514242872, 12.17139959950003, 12.95252978026018

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