Properties

Label 2-1110-185.159-c1-0-20
Degree $2$
Conductor $1110$
Sign $-0.763 + 0.645i$
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 + (−1.83 + 1.27i)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 + (2.22 − 0.191i)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 + (−0.820 + 0.572i)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 + (0.575 − 0.0494i)15-s + (−0.125 − 0.216i)16-s + (−0.191 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.763 + 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3869279726\)
\(L(\frac12)\) \(\approx\) \(0.3869279726\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 - 1.27i)T \)
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

−9.710054201336932262457614736072, −8.587639511350760465077517264633, −7.67667048413128027468277575700, −7.53568685625066292179670007796, −6.11625060614403351193770095716, −5.03332400675288867694185478977, −4.34661333125456480021053576875, −2.84314148595789477546177733820, −2.10183129288068606335353765952, −0.23414469533600582576276105574, 1.14018122864236500950864484947, 3.05932853896161479738697030414, 4.55877691911636275833670907458, 4.96404128521979409648223945403, 5.71259703938225904932020945578, 7.19941208521897590926996927740, 7.67087184707730983195253747855, 8.314338745714373440181672168019, 9.171473871607498169938883552444, 10.44870874095911328071658023321

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