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Properties

Label 2-1710-285.179-c1-0-37
Degree $2$
Conductor $1710$
Sign $-0.239 + 0.970i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.358 − 2.20i)5-s − 4.46i·7-s + 0.999i·8-s + (0.792 − 2.09i)10-s + 4.90i·11-s + (−2.12 − 3.67i)13-s + (2.23 − 3.86i)14-s + (−0.5 + 0.866i)16-s + (3.15 − 5.46i)17-s + (−3.5 + 2.59i)19-s + (1.73 − 1.41i)20-s + (−2.45 + 4.24i)22-s + (2.29 + 3.96i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.160 − 0.987i)5-s − 1.68i·7-s + 0.353i·8-s + (0.250 − 0.661i)10-s + 1.47i·11-s + (−0.588 − 1.01i)13-s + (0.596 − 1.03i)14-s + (−0.125 + 0.216i)16-s + (0.765 − 1.32i)17-s + (−0.802 + 0.596i)19-s + (0.387 − 0.316i)20-s + (−0.522 + 0.905i)22-s + (0.477 + 0.827i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.239 + 0.970i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.717542576\)
\(L(\frac12)\) \(\approx\) \(1.717542576\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.358 + 2.20i)T \)
19 \( 1 + (3.5 - 2.59i)T \)
good7 \( 1 + 4.46iT - 7T^{2} \)
11 \( 1 - 4.90iT - 11T^{2} \)
13 \( 1 + (2.12 + 3.67i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.15 + 5.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.29 - 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.67 + 6.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 7.73T + 37T^{2} \)
41 \( 1 + (-5.47 + 9.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.49 - 2.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.31 + 10.9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.67 - 0.968i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.22 - 2.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.468 - 0.811i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.12 - 3.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.67 + 6.36i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.85 + 5.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 + 3.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (-3.02 - 5.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.12 + 3.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.158242239340189258174830417939, −7.84924294540187504842415855036, −7.53891606311831704548987599331, −6.95211413804308443187585333959, −5.59350174467623055334454075165, −4.91694543067001279356239027888, −4.23116228828489532099135500136, −3.45265204625980133318700915664, −1.91580711248395424659105535561, −0.49969077604160881505230895068, 1.83361191679254322888254272162, 2.78579657623476819262387940381, 3.39579367804926990075149723753, 4.57626451741849583860056808917, 5.65536387709258928060330576257, 6.17790656998897392553629273357, 6.85867315007927667600771885071, 8.125784099098772298874918308143, 8.794840562043305212356550810999, 9.537622039837620271251480622392

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