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Properties

Label 2-1710-19.7-c1-0-5
Degree $2$
Conductor $1710$
Sign $-0.999 - 0.0349i$
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 + 3.84·7-s − 0.999·8-s + (−0.499 + 0.866i)10-s − 5.73·11-s + (−2.36 + 4.10i)13-s + (1.92 + 3.32i)14-s + (−0.5 − 0.866i)16-s + (−3.31 − 5.73i)17-s + (−4.29 + 0.771i)19-s − 0.999·20-s + (−2.86 − 4.96i)22-s + (−2.86 + 4.96i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 1.45·7-s − 0.353·8-s + (−0.158 + 0.273i)10-s − 1.72·11-s + (−0.656 + 1.13i)13-s + (0.513 + 0.889i)14-s + (−0.125 − 0.216i)16-s + (−0.803 − 1.39i)17-s + (−0.984 + 0.176i)19-s − 0.223·20-s + (−0.611 − 1.05i)22-s + (−0.597 + 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0349i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.999 - 0.0349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.205956923\)
\(L(\frac12)\) \(\approx\) \(1.205956923\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.29 - 0.771i)T \)
good7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 + 5.73T + 11T^{2} \)
13 \( 1 + (2.36 - 4.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.31 + 5.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.86 - 4.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.15 - 8.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-4.39 - 7.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.523 + 0.905i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.890 + 1.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.55 + 4.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0545 + 0.0945i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.94 - 5.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.31 + 5.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.05 - 5.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.36 - 5.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + (2.55 - 4.42i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.57 + 7.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.604369281879750351747418533963, −8.767600818519069993314176123040, −7.928373486263282600770914523393, −7.37786202634509624608759881795, −6.65537106446498333259531940959, −5.41557365202358964371978703994, −4.98459104588082775690697021182, −4.20233924723427914134508508438, −2.73404266841007367065820807432, −1.93273599126173492818687896023, 0.36321989711770345202330099405, 2.06007310842979834942074508299, 2.46749543500379366923860235979, 4.09195758075160160462586400485, 4.74157421075679348500260063276, 5.45979123514510359019657440298, 6.20996154168174895212127377719, 7.80342064900956285999221025870, 8.054768759217033374084297829672, 8.831267824072145317638167703111

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