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Properties

Label 2-930-31.4-c1-0-16
Degree $2$
Conductor $930$
Sign $-0.136 + 0.990i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 0.999·6-s + (3.33 − 2.42i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (3.83 − 2.78i)11-s + (0.309 + 0.951i)12-s + (−0.852 + 2.62i)13-s + (−3.33 − 2.42i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (1.49 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.178 − 0.549i)3-s + (−0.404 + 0.293i)4-s + 0.447·5-s − 0.408·6-s + (1.26 − 0.916i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (1.15 − 0.840i)11-s + (0.0892 + 0.274i)12-s + (−0.236 + 0.727i)13-s + (−0.891 − 0.647i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.362 + 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.136 + 0.990i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.136 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24647 - 1.43073i\)
\(L(\frac12)\) \(\approx\) \(1.24647 - 1.43073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 - T \)
31 \( 1 + (1.18 + 5.44i)T \)
good7 \( 1 + (-3.33 + 2.42i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-3.83 + 2.78i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.852 - 2.62i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.49 - 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.31 - 4.05i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.37 - 1.72i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.923 - 2.84i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 5.20T + 37T^{2} \)
41 \( 1 + (0.377 + 1.16i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.20 - 6.79i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.01 + 6.21i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.16 + 6.66i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.57 + 7.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 1.79T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (-6.34 - 4.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.97 - 7.24i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.85 + 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.676 - 2.08i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.36 + 3.16i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (11.6 - 8.47i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.856772532860386488580348526982, −9.014540475160104240972284732615, −8.242120994560546063327429407382, −7.46918859259170127858224995547, −6.53425530969901720883929070776, −5.40886320108503080268456432071, −4.25826198430826676978650250793, −3.39227902451549236760542551872, −1.80949227984915536831731923752, −1.17711940136578951935803177024, 1.52330599551716431718764276781, 2.84943512488228979232392794797, 4.44431954240138262395497839518, 5.05707352158440386442354098792, 5.85177517215608762021343648386, 6.98733187790681466441807025986, 7.81306391729839347948111174090, 8.885556441651490361467244145085, 9.115396451971809200281062386255, 10.12463689340650789500377613678

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