↑

Properties

Label 2-1110-37.11-c1-0-5
Degree $2$
Conductor $1110$
Sign $-0.727 - 0.686i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (−0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s − 0.999i·6-s + (1.35 − 2.35i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s − 3.71·11-s + (0.499 + 0.866i)12-s + (0.0894 + 0.0516i)13-s + 2.71i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (−4.85 + 2.80i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s − 0.408i·6-s + (0.513 − 0.889i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s − 1.12·11-s + (0.144 + 0.249i)12-s + (0.0248 + 0.0143i)13-s + 0.726i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (−1.17 + 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.727 - 0.686i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7162875334\)
\(L(\frac12)\) \(\approx\) \(0.7162875334\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-3.02 - 5.27i)T \)
good7 \( 1 + (-1.35 + 2.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 + (-0.0894 - 0.0516i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.85 - 2.80i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.581 + 0.335i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.33iT - 23T^{2} \)
29 \( 1 - 6.22iT - 29T^{2} \)
31 \( 1 - 5.43iT - 31T^{2} \)
41 \( 1 + (-4.16 + 7.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 - 0.896T + 47T^{2} \)
53 \( 1 + (-6.66 - 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.08 - 1.78i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.20 + 4.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.52 + 6.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.90 - 11.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + (11.3 + 6.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.96 + 5.13i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (15.9 - 9.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.98iT - 97T^{2} \)
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

−10.27363433825955796256130122465, −9.360041342703793611515917310971, −8.547862667511932464635935202709, −7.66879484941299470207035010538, −6.93207536297729793862120226359, −5.98033905613655389316755233479, −5.08120027974867054402534221674, −4.22836607411148338386206648193, −2.83631051751380630847316932244, −1.41493246290299568667428908358, 0.39879132026300667818361718472, 2.15832135797192790414888417909, 2.52849084527639835563666884929, 4.36650995120088015141400690109, 5.35822826519778233505493618361, 6.14678408599285146947151414747, 7.17720757568986085217448276263, 8.076320905035796755925240141452, 8.655914605348159295788038458937, 9.491029513864448970536159763882

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