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Properties

Label 2-1710-19.11-c1-0-33
Degree $2$
Conductor $1710$
Sign $-0.996 - 0.0791i$
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 − 1.44·7-s − 0.999·8-s + (−0.499 − 0.866i)10-s + 3·11-s + (−2.44 − 4.24i)13-s + (−0.724 + 1.25i)14-s + (−0.5 + 0.866i)16-s + (−0.775 + 1.34i)17-s + (−4.17 − 1.25i)19-s − 0.999·20-s + (1.5 − 2.59i)22-s + (−2.94 − 5.10i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 0.547·7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + 0.904·11-s + (−0.679 − 1.17i)13-s + (−0.193 + 0.335i)14-s + (−0.125 + 0.216i)16-s + (−0.188 + 0.325i)17-s + (−0.957 − 0.287i)19-s − 0.223·20-s + (0.319 − 0.553i)22-s + (−0.615 − 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.996 - 0.0791i$
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: \((2,\ 1710,\ (\ :1/2),\ -0.996 - 0.0791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.072747841\)
\(L(\frac12)\) \(\approx\) \(1.072747841\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.17 + 1.25i)T \)
good7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.775 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.94 + 5.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.67 - 8.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.898T + 31T^{2} \)
37 \( 1 + 9.89T + 37T^{2} \)
41 \( 1 + (-2.17 + 3.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.89 + 10.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.550 + 0.953i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.72 + 6.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.22 + 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.67 - 6.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.67 - 2.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.22 - 9.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.44 - 5.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (2.72 + 4.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.22 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.822037805460614096207944828423, −8.558414263826470734814951141551, −7.16433099574126084709265120496, −6.42600678054899802388618249895, −5.55275583292283422990931648909, −4.70324943426869559746225864765, −3.80543867876480440991342033604, −2.85823941048769197660856729955, −1.77739471220348028030001812770, −0.33919502694105713197746501599, 1.82672371183533862965876604855, 3.01130939676192473460891446401, 4.07765589974805824483946791460, 4.69669010811083059041692317003, 6.08827163356523936160804620498, 6.35809831672096645700710969645, 7.20768352563735627142786153360, 7.976716641393189526651048420278, 9.104972877750833538012141985413, 9.492731393390269898968099202464

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