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Properties

Label 2-1110-37.11-c1-0-12
Degree $2$
Conductor $1110$
Sign $0.989 - 0.141i$
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  + (−0.866 + 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 + (−0.267 + 0.463i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 5.44·11-s + (0.499 + 0.866i)12-s + (0.556 + 0.321i)13-s − 0.535i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−0.402 + 0.232i)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 + (−0.101 + 0.175i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 1.64·11-s + (0.144 + 0.249i)12-s + (0.154 + 0.0890i)13-s − 0.143i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.0975 + 0.0563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.989 - 0.141i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9821437173\)
\(L(\frac12)\) \(\approx\) \(0.9821437173\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.01 - 0.924i)T \)
good7 \( 1 + (0.267 - 0.463i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.44T + 11T^{2} \)
13 \( 1 + (-0.556 - 0.321i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.402 - 0.232i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.12 + 4.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.37iT - 23T^{2} \)
29 \( 1 + 5.38iT - 29T^{2} \)
31 \( 1 - 1.32iT - 31T^{2} \)
41 \( 1 + (-4.86 + 8.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.04iT - 43T^{2} \)
47 \( 1 - 8.14T + 47T^{2} \)
53 \( 1 + (-2.89 - 5.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.2 + 5.90i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.55 - 1.47i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.04 - 3.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.17 + 3.76i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.60T + 73T^{2} \)
79 \( 1 + (3.48 + 2.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.61 - 6.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.69 + 2.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.43iT - 97T^{2} \)
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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

−9.676453186363252106695995331172, −8.927247462919564685161037490604, −8.564547994829500946175974275225, −7.32051293131137478173818227392, −6.48759268983905653980216475712, −5.87624218784653048548305873217, −4.50299445739239789965533891413, −3.97177705035196863018588242533, −2.34802342547754545347732284620, −0.72863556826476402441704807177, 1.00945928396299996330359097864, 2.16174851244368822205487184355, 3.59146099562082538959491551060, 4.30637605054238017551528588731, 5.90437811156911238927859872440, 6.62250813856261648018082298654, 7.32454917053030871872077718148, 8.291055144085323685860202952235, 8.929445341544575506097800541710, 9.856174490064665310033488130148

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