Properties

Label 2-327990-1.1-c1-0-24
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 + 5·7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s + 13-s − 5·14-s − 15-s + 16-s − 3·17-s − 18-s − 4·19-s + 20-s − 5·21-s + 4·22-s − 3·23-s + 24-s + 25-s − 26-s − 27-s + 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.88·7-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 − 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.09·21-s + 0.852·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.944·28-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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−12.68954514648604, −12.36507147628141, −11.57868405761026, −11.49990490866750, −10.87709439336333, −10.63289295553686, −10.25584843875251, −9.803177535530621, −8.980645916814819, −8.708731635997788, −8.180549620458447, −7.918030624053488, −7.399664076101027, −6.748071489719074, −6.462631279550198, −5.664472516957725, −5.300970974997222, −5.021272864435912, −4.330900063271195, −3.923220402703796, −2.998775207143953, −2.239829427867945, −1.976224050227952, −1.510154606472509, −0.7086887690947975, 0, 0.7086887690947975, 1.510154606472509, 1.976224050227952, 2.239829427867945, 2.998775207143953, 3.923220402703796, 4.330900063271195, 5.021272864435912, 5.300970974997222, 5.664472516957725, 6.462631279550198, 6.748071489719074, 7.399664076101027, 7.918030624053488, 8.180549620458447, 8.708731635997788, 8.980645916814819, 9.803177535530621, 10.25584843875251, 10.63289295553686, 10.87709439336333, 11.49990490866750, 11.57868405761026, 12.36507147628141, 12.68954514648604

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