Properties

Label 2-327990-1.1-c1-0-19
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 − 2·7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 6·17-s + 18-s + 2·19-s − 20-s − 2·21-s + 2·22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-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 − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-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

\(L(1)\) \(\approx\) \(6.199055062\)
\(L(\frac12)\) \(\approx\) \(6.199055062\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.48533513829559, −12.17282405627865, −11.98061859406727, −11.42504686995808, −10.68535234373047, −10.41572599712519, −9.798236508263972, −9.528339798961329, −8.939615225545060, −8.427584233014657, −7.804196182457393, −7.591187911874728, −7.105211398307788, −6.339722614783691, −6.097135953774386, −5.719623029890810, −4.835816943756646, −4.490003617470602, −3.907922046302105, −3.364434031383319, −3.201005088613849, −2.523977383249229, −1.855651664799961, −1.140344759654117, −0.5972779218646426, 0.5972779218646426, 1.140344759654117, 1.855651664799961, 2.523977383249229, 3.201005088613849, 3.364434031383319, 3.907922046302105, 4.490003617470602, 4.835816943756646, 5.719623029890810, 6.097135953774386, 6.339722614783691, 7.105211398307788, 7.591187911874728, 7.804196182457393, 8.427584233014657, 8.939615225545060, 9.528339798961329, 9.798236508263972, 10.41572599712519, 10.68535234373047, 11.42504686995808, 11.98061859406727, 12.17282405627865, 12.48533513829559

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