Properties

Label 2-2790-1.1-c1-0-44
Degree $2$
Conductor $2790$
Sign $-1$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
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 + 4-s − 5-s − 2·7-s + 8-s − 10-s + 4·11-s − 4·13-s − 2·14-s + 16-s − 2·17-s − 8·19-s − 20-s + 4·22-s + 8·23-s + 25-s − 4·26-s − 2·28-s − 4·29-s − 31-s + 32-s − 2·34-s + 2·35-s − 12·37-s − 8·38-s − 40-s − 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.377·28-s − 0.742·29-s − 0.179·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s − 1.97·37-s − 1.29·38-s − 0.158·40-s − 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $-1$
Analytic conductor: \(22.2782\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2790} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
31 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.597708845876214561601504797981, −7.27247978705351858604903144155, −6.89339043212249499085076241028, −6.24527763543800026157189775230, −5.18702367972907479590415828598, −4.40383834475218934536745920625, −3.69311022997520404123807155694, −2.82884119268115589326918618685, −1.73336120153618990694100542941, 0, 1.73336120153618990694100542941, 2.82884119268115589326918618685, 3.69311022997520404123807155694, 4.40383834475218934536745920625, 5.18702367972907479590415828598, 6.24527763543800026157189775230, 6.89339043212249499085076241028, 7.27247978705351858604903144155, 8.597708845876214561601504797981

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