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Properties

Label 2-1710-95.49-c1-0-11
Degree $2$
Conductor $1710$
Sign $-0.541 - 0.840i$
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 + (−0.384 − 2.20i)5-s + 2.51i·7-s + 0.999i·8-s + (0.768 − 2.09i)10-s − 2.88·11-s + (−4.03 + 2.32i)13-s + (−1.25 + 2.18i)14-s + (−0.5 + 0.866i)16-s + (6.31 + 3.64i)17-s + (4.11 + 1.43i)19-s + (1.71 − 1.43i)20-s + (−2.49 − 1.44i)22-s + (−1.48 + 0.855i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.171 − 0.985i)5-s + 0.951i·7-s + 0.353i·8-s + (0.242 − 0.664i)10-s − 0.869·11-s + (−1.11 + 0.645i)13-s + (−0.336 + 0.582i)14-s + (−0.125 + 0.216i)16-s + (1.53 + 0.883i)17-s + (0.943 + 0.330i)19-s + (0.383 − 0.320i)20-s + (−0.532 − 0.307i)22-s + (−0.309 + 0.178i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.588277913\)
\(L(\frac12)\) \(\approx\) \(1.588277913\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.384 + 2.20i)T \)
19 \( 1 + (-4.11 - 1.43i)T \)
good7 \( 1 - 2.51iT - 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
13 \( 1 + (4.03 - 2.32i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.31 - 3.64i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.48 - 0.855i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.20 + 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.22T + 31T^{2} \)
37 \( 1 - 3.26iT - 37T^{2} \)
41 \( 1 + (4.48 - 7.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.84 - 3.37i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.98 + 2.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.38 - 5.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.73 - 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.50 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.23 - 5.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.38 + 9.31i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.2 - 7.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.11 + 7.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.14iT - 83T^{2} \)
89 \( 1 + (2.37 + 4.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.98 - 1.14i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.501469795956553713097157548881, −8.750118622539867106988377193864, −7.71965623931182687060106023168, −7.56955354822348963171735851801, −5.99581197720267949818704906895, −5.52179063489856507602548979636, −4.86961089836424487208446086862, −3.86317059531303692910233247242, −2.78201863253081374038629709945, −1.62834888384198467023632619175, 0.46519385670157004278044076451, 2.20088023728568308376676595858, 3.19603107990726547272839299921, 3.71208766431895393906988617773, 5.13760263032431883592146937503, 5.45520429856907352431690960152, 6.82884807744529818099774683059, 7.50555442644424826382051132922, 7.75821870405251053852585293063, 9.496186940907052014155380552122

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