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Properties

Label 2-1110-185.64-c1-0-6
Degree $2$
Conductor $1110$
Sign $0.00653 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.191 + 2.22i)5-s − 0.999i·6-s + (−2.57 + 1.48i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (2.02 + 0.948i)10-s − 6.26·11-s + (−0.866 − 0.499i)12-s + (1.50 + 2.60i)13-s + 2.96i·14-s + (1.27 + 1.83i)15-s + (−0.5 + 0.866i)16-s + (0.790 − 1.36i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.0856 + 0.996i)5-s − 0.408i·6-s + (−0.971 + 0.560i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.640 + 0.299i)10-s − 1.88·11-s + (−0.249 − 0.144i)12-s + (0.417 + 0.723i)13-s + 0.793i·14-s + (0.330 + 0.473i)15-s + (−0.125 + 0.216i)16-s + (0.191 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00653 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00653 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.00653 - 0.999i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.00653 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9773554298\)
\(L(\frac12)\) \(\approx\) \(0.9773554298\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.191 - 2.22i)T \)
37 \( 1 + (4.55 - 4.03i)T \)
good7 \( 1 + (2.57 - 1.48i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 6.26T + 11T^{2} \)
13 \( 1 + (-1.50 - 2.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.790 + 1.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.22 + 0.708i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.48T + 23T^{2} \)
29 \( 1 - 7.54iT - 29T^{2} \)
31 \( 1 - 7.23iT - 31T^{2} \)
41 \( 1 + (-2.98 - 5.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.168T + 43T^{2} \)
47 \( 1 + 3.32iT - 47T^{2} \)
53 \( 1 + (0.224 + 0.129i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.62 + 2.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.53 - 4.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.1 + 7.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.03 + 5.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.77iT - 73T^{2} \)
79 \( 1 + (2.15 - 1.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.43 + 4.29i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (10.6 + 6.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.84T + 97T^{2} \)
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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

−10.18448837494217924376875318570, −9.389610251383168029889793564471, −8.518437771750720396407945824759, −7.48209060277959418938606678827, −6.68013270471427189150921228741, −5.83130353447230642386361624085, −4.83946170818649691646655456444, −3.24886309468042457801135784872, −3.02455118820082768369332722973, −1.93406382531776958715593601863, 0.33051065615745944591266890375, 2.44856754507019209839591917122, 3.57357930606806909123220737098, 4.38683640860071402003988982077, 5.48924544263939241209505089019, 5.95989049201501750804718389338, 7.40430868962103955894061165328, 7.955080181893871806737874469028, 8.592016178475219467605107206639, 9.750385223502947899706284815279

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