Properties

Label 2-327990-1.1-c1-0-20
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 − 4·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 13-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 20-s − 4·21-s − 4·22-s + 4·23-s − 24-s + 25-s − 26-s + 27-s − 4·28-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-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 + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.872·21-s − 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s − 0.182·30-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

\(L(1)\) \(\approx\) \(3.149185112\)
\(L(\frac12)\) \(\approx\) \(3.149185112\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.84553327858890, −12.14167403631781, −11.69280799309404, −11.05751951208302, −10.81939751854507, −10.02326915251876, −9.726010387578022, −9.491148798951124, −8.943824770520645, −8.770419331924342, −8.061660523749179, −7.455505587582886, −7.052538984974219, −6.665514753462028, −6.157630416300274, −5.779775121381421, −5.216685578200456, −4.258203506063068, −3.881381981326448, −3.368761190721891, −2.867717351924257, −2.361879797067125, −1.711338234366580, −0.9736025040462348, −0.5998711734440186, 0.5998711734440186, 0.9736025040462348, 1.711338234366580, 2.361879797067125, 2.867717351924257, 3.368761190721891, 3.881381981326448, 4.258203506063068, 5.216685578200456, 5.779775121381421, 6.157630416300274, 6.665514753462028, 7.052538984974219, 7.455505587582886, 8.061660523749179, 8.770419331924342, 8.943824770520645, 9.491148798951124, 9.726010387578022, 10.02326915251876, 10.81939751854507, 11.05751951208302, 11.69280799309404, 12.14167403631781, 12.84553327858890

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