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Properties

Label 2-1110-37.10-c1-0-9
Degree $2$
Conductor $1110$
Sign $-0.744 - 0.667i$
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 + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (1.74 + 3.02i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s + 3.40·11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)13-s − 3.49·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−3.24 + 5.62i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (0.660 + 1.14i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s + 1.02·11-s + (0.144 − 0.249i)12-s + (−0.138 − 0.240i)13-s − 0.934·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.787 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594963728\)
\(L(\frac12)\) \(\approx\) \(1.594963728\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (5.99 - 1.02i)T \)
good7 \( 1 + (-1.74 - 3.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.40T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.24 - 5.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.20 + 2.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.183T + 23T^{2} \)
29 \( 1 - 8.71T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
41 \( 1 + (2.85 + 4.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.99T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 + (3.94 - 6.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.19 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.454 + 0.786i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.65 + 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.83 - 4.91i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.80T + 73T^{2} \)
79 \( 1 + (8.58 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.34 - 4.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.495 + 0.858i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.15267249921925665497507244212, −8.950496562603571772882412396580, −8.727199135167230231472163141710, −7.926199330033924228239308623523, −6.61015026410854806368501651679, −6.15736688298733304027937210612, −5.03252973761139236973692498950, −4.25070621057293432304899957276, −2.85341093169782122491393839356, −1.69984543007247144754794096495, 0.822427677916467684080675468323, 1.74681179783579770928297563728, 3.02418391650948360160706753043, 4.26966253886025470497031721016, 4.84051806276889265316204692287, 6.51794242361370534955206365589, 7.03368073180033323679202983277, 8.136942784074119347464460925829, 8.620711085575949249906006946651, 9.628138496193702932086802385741

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