Properties

Label 2-327990-1.1-c1-0-9
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 + 4·7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 13-s − 4·14-s − 15-s + 16-s − 4·17-s − 18-s + 20-s − 4·21-s − 4·22-s − 2·23-s + 24-s + 25-s + 26-s − 27-s + 4·28-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 + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.223·20-s − 0.872·21-s − 0.852·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-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

\(L(1)\) \(\approx\) \(2.150721427\)
\(L(\frac12)\) \(\approx\) \(2.150721427\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.35391187369406, −11.85674102368069, −11.71285595632545, −11.27432816744417, −10.77066971121665, −10.41045995463133, −9.870880323052038, −9.418408690902812, −8.905008250710712, −8.552872266332608, −8.003278722817030, −7.642089440581846, −6.922302484980960, −6.635736073526177, −6.163323652912343, −5.644762207407059, −4.879046255608127, −4.778029602669070, −4.098520450335271, −3.531980120550937, −2.674749475814273, −2.013774753600058, −1.695360089817415, −1.143347357218842, −0.4755259453450882, 0.4755259453450882, 1.143347357218842, 1.695360089817415, 2.013774753600058, 2.674749475814273, 3.531980120550937, 4.098520450335271, 4.778029602669070, 4.879046255608127, 5.644762207407059, 6.163323652912343, 6.635736073526177, 6.922302484980960, 7.642089440581846, 8.003278722817030, 8.552872266332608, 8.905008250710712, 9.418408690902812, 9.870880323052038, 10.41045995463133, 10.77066971121665, 11.27432816744417, 11.71285595632545, 11.85674102368069, 12.35391187369406

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