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Properties

Label 2-1710-95.18-c1-0-7
Degree $2$
Conductor $1710$
Sign $0.336 - 0.941i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.253 − 2.22i)5-s + (−2.47 + 2.47i)7-s + (0.707 − 0.707i)8-s + (−1.75 + 1.39i)10-s − 2.74·11-s + (−1.20 + 1.20i)13-s + 3.50·14-s − 1.00·16-s + (4.87 − 4.87i)17-s + (1.94 − 3.90i)19-s + (2.22 + 0.253i)20-s + (1.94 + 1.94i)22-s + (−0.0321 − 0.0321i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.113 − 0.993i)5-s + (−0.935 + 0.935i)7-s + (0.250 − 0.250i)8-s + (−0.553 + 0.440i)10-s − 0.827·11-s + (−0.333 + 0.333i)13-s + 0.935·14-s − 0.250·16-s + (1.18 − 1.18i)17-s + (0.445 − 0.895i)19-s + (0.496 + 0.0567i)20-s + (0.413 + 0.413i)22-s + (−0.00670 − 0.00670i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.336 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5311124863\)
\(L(\frac12)\) \(\approx\) \(0.5311124863\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.253 + 2.22i)T \)
19 \( 1 + (-1.94 + 3.90i)T \)
good7 \( 1 + (2.47 - 2.47i)T - 7iT^{2} \)
11 \( 1 + 2.74T + 11T^{2} \)
13 \( 1 + (1.20 - 1.20i)T - 13iT^{2} \)
17 \( 1 + (-4.87 + 4.87i)T - 17iT^{2} \)
23 \( 1 + (0.0321 + 0.0321i)T + 23iT^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 - 6.50iT - 31T^{2} \)
37 \( 1 + (-4.58 - 4.58i)T + 37iT^{2} \)
41 \( 1 - 5.96iT - 41T^{2} \)
43 \( 1 + (-5.39 - 5.39i)T + 43iT^{2} \)
47 \( 1 + (3.66 - 3.66i)T - 47iT^{2} \)
53 \( 1 + (8.97 - 8.97i)T - 53iT^{2} \)
59 \( 1 + 4.42T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 + (-7.00 - 7.00i)T + 67iT^{2} \)
71 \( 1 - 5.56iT - 71T^{2} \)
73 \( 1 + (2.19 + 2.19i)T + 73iT^{2} \)
79 \( 1 + 0.225T + 79T^{2} \)
83 \( 1 + (-3.87 - 3.87i)T + 83iT^{2} \)
89 \( 1 - 9.13T + 89T^{2} \)
97 \( 1 + (8.76 + 8.76i)T + 97iT^{2} \)
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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

−9.505457401267778692538280667853, −9.002064371735542428379313689304, −8.010347864989213997288663269066, −7.37172664270770046147912067042, −6.20231212618105239524631445964, −5.30802338966611563006993060527, −4.64966698242989988496470952002, −3.17038668772126261198469116718, −2.59236972095033699480883263520, −1.15412793610730817328087623295, 0.25867940434603369271400333870, 1.97946122212877075278472264978, 3.29433436368631174260345594038, 3.91209261407000987826122878334, 5.51252401155409274457023604784, 6.00491457803426652415160888466, 6.91750983565398802329298466498, 7.71678553315409599496213299460, 7.940589549812783649421965591410, 9.477594089150994289051193435726

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