Properties

Label 2-1290-1.1-c1-0-1
Degree $2$
Conductor $1290$
Sign $1$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
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 + 4·13-s − 4·14-s + 15-s + 16-s + 4·17-s + 18-s + 4·19-s − 20-s + 4·21-s − 4·22-s + 8·23-s − 24-s + 25-s + 4·26-s − 27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1290\)    =    \(2 \cdot 3 \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1290} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1290,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.656216905\)
\(L(\frac12)\) \(\approx\) \(1.656216905\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.839327328437397982778167885587, −8.932207006890475896112577273389, −7.72246877755849266974156581562, −7.09567091263591639141288364980, −6.13728646627375840423934830636, −5.57282709982714713057594863606, −4.59719533120211205047653684382, −3.38729842608415021875260061806, −2.93213511092676663916020642413, −0.883863689030719145520692822605, 0.883863689030719145520692822605, 2.93213511092676663916020642413, 3.38729842608415021875260061806, 4.59719533120211205047653684382, 5.57282709982714713057594863606, 6.13728646627375840423934830636, 7.09567091263591639141288364980, 7.72246877755849266974156581562, 8.932207006890475896112577273389, 9.839327328437397982778167885587

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