↑

Properties

Label 2-1110-185.184-c1-0-11
Degree $2$
Conductor $1110$
Sign $0.583 - 0.812i$
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  + 2-s − i·3-s + 4-s + (0.718 + 2.11i)5-s − i·6-s + 2.74i·7-s + 8-s − 9-s + (0.718 + 2.11i)10-s − 0.572·11-s − i·12-s − 3.04·13-s + 2.74i·14-s + (2.11 − 0.718i)15-s + 16-s + 4.95·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.321 + 0.946i)5-s − 0.408i·6-s + 1.03i·7-s + 0.353·8-s − 0.333·9-s + (0.227 + 0.669i)10-s − 0.172·11-s − 0.288i·12-s − 0.843·13-s + 0.734i·14-s + (0.546 − 0.185i)15-s + 0.250·16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.583 - 0.812i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.583 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.486209171\)
\(L(\frac12)\) \(\approx\) \(2.486209171\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-0.718 - 2.11i)T \)
37 \( 1 + (-4.94 + 3.53i)T \)
good7 \( 1 - 2.74iT - 7T^{2} \)
11 \( 1 + 0.572T + 11T^{2} \)
13 \( 1 + 3.04T + 13T^{2} \)
17 \( 1 - 4.95T + 17T^{2} \)
19 \( 1 - 3.74iT - 19T^{2} \)
23 \( 1 - 2.98T + 23T^{2} \)
29 \( 1 - 7.36iT - 29T^{2} \)
31 \( 1 - 1.12iT - 31T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 6.91T + 43T^{2} \)
47 \( 1 - 0.776iT - 47T^{2} \)
53 \( 1 - 3.55iT - 53T^{2} \)
59 \( 1 + 9.27iT - 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 - 9.09iT - 67T^{2} \)
71 \( 1 - 6.26T + 71T^{2} \)
73 \( 1 + 7.83iT - 73T^{2} \)
79 \( 1 + 0.716iT - 79T^{2} \)
83 \( 1 + 2.01iT - 83T^{2} \)
89 \( 1 - 8.33iT - 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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

−10.08268113490854388756551177245, −9.214992303932901757534761028348, −8.058748866455888572966258185233, −7.32813605308094154247037458069, −6.53316836443426093584036154792, −5.68055152661175846041187740302, −5.11213493759976719364032591656, −3.47352137880760969150356758742, −2.74687113923873340346084095456, −1.77598788969883523576702958476, 0.889148165910405021091991329025, 2.51385970633407331468132211132, 3.72441299074186318254033824091, 4.59872429274313583347101710458, 5.15509038376395190646552775951, 6.10220740026771661514765477748, 7.24990166409057472214810140154, 7.949725677494959137645024473514, 9.027762688130413033217317123392, 9.943028354212606269491138554255

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