Properties

Label 2-327990-1.1-c1-0-40
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 − 2·7-s + 8-s + 9-s − 10-s + 3·11-s + 12-s − 13-s − 2·14-s − 15-s + 16-s + 7·17-s + 18-s + 2·19-s − 20-s − 2·21-s + 3·22-s + 4·23-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 + 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s + 0.639·22-s + 0.834·23-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−12.87654454212185, −12.38523184419974, −12.02693021535192, −11.56644635164068, −11.28470049715776, −10.37673294374494, −10.12972305892593, −9.759743705490958, −9.081297463528161, −8.789946798015994, −8.217161933367279, −7.543932053975941, −7.284119753336645, −6.852443278650323, −6.357782564239852, −5.657137159877334, −5.354029481870128, −4.768421165429895, −4.045909959916273, −3.678402389332015, −3.331038516468899, −2.832649714259353, −2.252239751293578, −1.338208156604411, −1.051424249141863, 0, 1.051424249141863, 1.338208156604411, 2.252239751293578, 2.832649714259353, 3.331038516468899, 3.678402389332015, 4.045909959916273, 4.768421165429895, 5.354029481870128, 5.657137159877334, 6.357782564239852, 6.852443278650323, 7.284119753336645, 7.543932053975941, 8.217161933367279, 8.789946798015994, 9.081297463528161, 9.759743705490958, 10.12972305892593, 10.37673294374494, 11.28470049715776, 11.56644635164068, 12.02693021535192, 12.38523184419974, 12.87654454212185

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