Properties

Label 2-327990-1.1-c1-0-14
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 − 2·11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 3·17-s − 18-s − 6·19-s + 20-s − 21-s + 2·22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-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 − 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-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 + ⋯

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

\(L(1)\) \(\approx\) \(1.918298041\)
\(L(\frac12)\) \(\approx\) \(1.918298041\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−12.35652506188294, −12.30026719584762, −11.59800278466475, −11.00812374655989, −10.83868414964196, −10.28282022442633, −10.05464214118558, −9.385001514576981, −9.038582481653637, −8.350260622783140, −8.099335613187831, −7.605571978725327, −7.045919759062916, −6.520135202173922, −6.015114874270576, −5.803529111386027, −5.022945060233129, −4.707648809221584, −4.043284367724024, −3.415667303644427, −2.734778012632047, −2.141033720105392, −1.759918919443075, −0.8954028508383895, −0.5249163316582929, 0.5249163316582929, 0.8954028508383895, 1.759918919443075, 2.141033720105392, 2.734778012632047, 3.415667303644427, 4.043284367724024, 4.707648809221584, 5.022945060233129, 5.803529111386027, 6.015114874270576, 6.520135202173922, 7.045919759062916, 7.605571978725327, 8.099335613187831, 8.350260622783140, 9.038582481653637, 9.385001514576981, 10.05464214118558, 10.28282022442633, 10.83868414964196, 11.00812374655989, 11.59800278466475, 12.30026719584762, 12.35652506188294

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