Properties

Label 2-1710-19.7-c1-0-7
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 − 2·11-s + (1.5 − 2.59i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (2 + 3.46i)17-s + (4 + 1.73i)19-s + 0.999·20-s + (−1 − 1.73i)22-s + (−3 + 5.19i)23-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 − 0.603·11-s + (0.416 − 0.720i)13-s + (−0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.485 + 0.840i)17-s + (0.917 + 0.397i)19-s + 0.223·20-s + (−0.213 − 0.369i)22-s + (−0.625 + 1.08i)23-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: $\chi_{1710} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.140045759\)
\(L(\frac12)\) \(\approx\) \(1.140045759\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.521019738652887076679875482977, −8.735609011327007488369907528226, −7.79384514250086676435846351132, −7.52902298091379398069279549280, −6.26328222549425537948798407558, −5.61137450953318509240682647182, −4.93748198547746914391924650948, −3.69220654211477199101660299721, −3.16333239820633422659701529038, −1.43265587296690690164786913580, 0.38710904359593117618259384021, 2.04172403616598021533546531893, 2.97854462964633525383075663435, 3.82786039642966335783651288815, 4.76674731270656548056632891616, 5.68422017708166314169812374662, 6.54461899311020367173863015587, 7.39346916233407878148984357346, 8.248456154312129889804315922196, 9.263298283194245330597136393054

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