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Properties

Label 2-1710-19.7-c1-0-22
Degree $2$
Conductor $1710$
Sign $-0.571 + 0.820i$
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 − 2.64·7-s + 0.999·8-s + (−0.499 + 0.866i)10-s + 6.29·11-s + (1.32 + 2.29i)14-s + (−0.5 − 0.866i)16-s + (−2.82 − 4.88i)17-s + (−1.67 + 4.02i)19-s + 0.999·20-s + (−3.14 − 5.44i)22-s + (0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.999·7-s + 0.353·8-s + (−0.158 + 0.273i)10-s + 1.89·11-s + (0.353 + 0.612i)14-s + (−0.125 − 0.216i)16-s + (−0.684 − 1.18i)17-s + (−0.384 + 0.923i)19-s + 0.223·20-s + (−0.670 − 1.16i)22-s + (0.104 − 0.180i)23-s + (−0.0999 + 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.571 + 0.820i$
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: \((2,\ 1710,\ (\ :1/2),\ -0.571 + 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9865736404\)
\(L(\frac12)\) \(\approx\) \(0.9865736404\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 4.02i)T \)
good7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 - 6.29T + 11T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.82 + 4.88i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.82 - 3.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.29T + 31T^{2} \)
37 \( 1 - 8.29T + 37T^{2} \)
41 \( 1 + (4.32 + 7.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.29 + 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.32 - 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.64 - 4.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.46 - 6.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.82 + 8.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.82 + 11.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.82 + 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.64 + 4.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 + (-6.61 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.822 - 1.42i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.961474037219864716335764748140, −8.767720495093135292089741873853, −7.44794818849150460703986240972, −6.72577736397018625059187475443, −5.96673320891613018333237527336, −4.61836490173684225518573037464, −3.89290826962961152048744682599, −3.05651261851571669559143488205, −1.75431093194412928127002883335, −0.48420324971977805344032986611, 1.19414807944471293938863253447, 2.72326563602167566995223789701, 3.90559310176883780213941078030, 4.50542126701801813165472964284, 6.10571648487313531490543619817, 6.40929609977103996884990961679, 6.98468371528033808160711402824, 8.099759737346168736098810106969, 8.801467460541776288079636704010, 9.582357445534937367112716590356

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