↑

Properties

Label 2-1274-91.81-c1-0-24
Degree $2$
Conductor $1274$
Sign $-0.362 + 0.931i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s − 2·3-s + (−0.499 − 0.866i)4-s + (1.5 + 2.59i)5-s + (−1 + 1.73i)6-s − 0.999·8-s + 9-s + 3·10-s − 6·11-s + (0.999 + 1.73i)12-s + (2.5 − 2.59i)13-s + (−3 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)18-s − 4·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 1.15·3-s + (−0.249 − 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.408 + 0.707i)6-s − 0.353·8-s + 0.333·9-s + 0.948·10-s − 1.80·11-s + (0.288 + 0.499i)12-s + (0.693 − 0.720i)13-s + (−0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.117 − 0.204i)18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.362 + 0.931i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.362 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8797348840\)
\(L(\frac12)\) \(\approx\) \(0.8797348840\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
13 \( 1 + (-2.5 + 2.59i)T \)
good3 \( 1 + 2T + 3T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13024030832805194682509369085, −8.639936072485490303310731411999, −7.72624210244678318783533358724, −6.54011561890925544509267693486, −5.92040668065910547194774321977, −5.42570068989101336336322594401, −4.31333395414631523670745248383, −2.95277463893680232838264061508, −2.29854204199883681144386008912, −0.41711704599341553779825538564, 1.18558879339148341046448434604, 2.83210002057529430977321336610, 4.42779030613643452004089990310, 5.19244923480055986102350472147, 5.50979971400333567476390193774, 6.36448703817638744638362865128, 7.30837620136952892126107083068, 8.338909249566039839570749838659, 8.963535923003339267261030338877, 9.920460407297155320882558912263

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