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Properties

Label 2-1710-19.11-c1-0-21
Degree $2$
Conductor $1710$
Sign $0.371 + 0.928i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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 + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 1.75·7-s − 0.999·8-s + (−0.499 − 0.866i)10-s + 2.20·11-s + (1.60 + 2.77i)13-s + (0.876 − 1.51i)14-s + (−0.5 + 0.866i)16-s + (3.58 − 6.21i)17-s + (0.727 + 4.29i)19-s − 0.999·20-s + (1.10 − 1.91i)22-s + (1.10 + 1.91i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.662·7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + 0.666·11-s + (0.445 + 0.770i)13-s + (0.234 − 0.405i)14-s + (−0.125 + 0.216i)16-s + (0.869 − 1.50i)17-s + (0.166 + 0.985i)19-s − 0.223·20-s + (0.235 − 0.407i)22-s + (0.230 + 0.398i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.486678038\)
\(L(\frac12)\) \(\approx\) \(2.486678038\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.727 - 4.29i)T \)
good7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 - 2.20T + 11T^{2} \)
13 \( 1 + (-1.60 - 2.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.58 + 6.21i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.10 - 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.83 - 6.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.20T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-5.23 + 9.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.35 + 2.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.96 + 8.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.48 - 9.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.98 - 5.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0187 + 0.0324i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.58 + 6.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.98 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.604 - 1.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + (5.48 + 9.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.62 - 9.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.193525203181376443049790765635, −8.678387931562304171291366664528, −7.57882323051372268437624925787, −6.76886315992997769420508176758, −5.62988953938534921481882551822, −5.07450974015704032607332784407, −4.10466216660296874382364951943, −3.26256506819235183602423474582, −1.93163863113507336132041651264, −1.07441655089370514084270165518, 1.22506868499457810535065705296, 2.67955361423809943975507805962, 3.71561918262381216667853377636, 4.56836485964691076006774213762, 5.56831224940395947380001327969, 6.22599734156460221936400972023, 6.96995205511088219299552467994, 8.081901956210477261832420720742, 8.289785466929531929136904072200, 9.446398267017284756795565435721

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