Properties

Label 2-327990-1.1-c1-0-35
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 + 8-s + 9-s − 10-s + 12-s − 13-s − 15-s + 16-s + 6·17-s + 18-s − 20-s − 4·23-s + 24-s + 25-s − 26-s + 27-s − 30-s + 32-s + 6·34-s + 36-s + 6·37-s − 39-s − 40-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.182·30-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.160·39-s − 0.158·40-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.86657336675851, −12.33439895854158, −12.07606289089365, −11.65643403968043, −10.95855112121713, −10.74270905226299, −10.04019433871408, −9.578600541275313, −9.408729248989445, −8.509535726071079, −8.091739435961584, −7.768705937604711, −7.409922298586990, −6.731901844613836, −6.292732289434186, −5.798166794053443, −5.131046715257336, −4.856902975712537, −4.085778898464473, −3.792623524563603, −3.260794353244919, −2.743203904261683, −2.230811694417755, −1.499116489428893, −0.9521092752723906, 0, 0.9521092752723906, 1.499116489428893, 2.230811694417755, 2.743203904261683, 3.260794353244919, 3.792623524563603, 4.085778898464473, 4.856902975712537, 5.131046715257336, 5.798166794053443, 6.292732289434186, 6.731901844613836, 7.409922298586990, 7.768705937604711, 8.091739435961584, 8.509535726071079, 9.408729248989445, 9.578600541275313, 10.04019433871408, 10.74270905226299, 10.95855112121713, 11.65643403968043, 12.07606289089365, 12.33439895854158, 12.86657336675851

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