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Properties

Label 2-1710-19.11-c1-0-12
Degree $2$
Conductor $1710$
Sign $0.813 - 0.582i$
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 − 7-s + 0.999·8-s + (−0.499 − 0.866i)10-s − 6·11-s + (−2.5 − 4.33i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (4 − 1.73i)19-s + 0.999·20-s + (3 − 5.19i)22-s + (3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + 5·26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.377·7-s + 0.353·8-s + (−0.158 − 0.273i)10-s − 1.80·11-s + (−0.693 − 1.20i)13-s + (0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.917 − 0.397i)19-s + 0.223·20-s + (0.639 − 1.10i)22-s + (0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + 0.980·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.813 - 0.582i$
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: \((2,\ 1710,\ (\ :1/2),\ 0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9645323098\)
\(L(\frac12)\) \(\approx\) \(0.9645323098\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + 1.73i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.523171109724453074766595188422, −8.406408255314473290418651217003, −7.59449629469905035028858737949, −7.40450331673685842392819858653, −6.21191097719726046694913104037, −5.35310012251550029561371903598, −4.81600566699623103743042237138, −3.23123302787833580589793765112, −2.61796221773856478868216534150, −0.67110736924565267610648496899, 0.71703141297952087308926020841, 2.32481392672750532565795428611, 2.95398500532572820501256108502, 4.33184466067698072620885736122, 4.88150868462089107770567443747, 5.99418914435065781625643780136, 7.07509273453697507673996152864, 7.88089015461384895756798211898, 8.424494293813770971897252856811, 9.531831806814563650080064999592

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