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Properties

Label 2-1710-285.164-c1-0-32
Degree $2$
Conductor $1710$
Sign $0.370 + 0.928i$
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 + (2.09 + 0.792i)5-s + 2.01i·7-s − 0.999i·8-s + (2.20 − 0.358i)10-s − 6.31i·11-s + (2.12 − 3.67i)13-s + (1.00 + 1.74i)14-s + (−0.5 − 0.866i)16-s + (−1.42 − 2.46i)17-s + (−3.5 − 2.59i)19-s + (1.73 − 1.41i)20-s + (−3.15 − 5.47i)22-s + (−2.29 + 3.96i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.935 + 0.354i)5-s + 0.761i·7-s − 0.353i·8-s + (0.697 − 0.113i)10-s − 1.90i·11-s + (0.588 − 1.01i)13-s + (0.269 + 0.466i)14-s + (−0.125 − 0.216i)16-s + (−0.345 − 0.598i)17-s + (−0.802 − 0.596i)19-s + (0.387 − 0.316i)20-s + (−0.673 − 1.16i)22-s + (−0.477 + 0.827i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.370 + 0.928i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.868923218\)
\(L(\frac12)\) \(\approx\) \(2.868923218\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 - 0.792i)T \)
19 \( 1 + (3.5 + 2.59i)T \)
good7 \( 1 - 2.01iT - 7T^{2} \)
11 \( 1 + 6.31iT - 11T^{2} \)
13 \( 1 + (-2.12 + 3.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.42 + 2.46i)T + (-8.5 + 14.7i)T^{2} \)
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 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 + (-4.24 - 7.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.73 + 4.46i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.85 + 4.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.0 - 6.96i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.22 - 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.46 - 12.9i)T + (-30.5 - 52.8i)T^{2} \)
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 \( 1 + (1.36 - 0.790i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 - 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (-6.69 + 11.6i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.025685930511767484623114980098, −8.713773258033686573229214363664, −7.54985509627383644851042792307, −6.31131218793126478064552831877, −5.86481094211905382630564521652, −5.37792887840814521661658640674, −4.04288948257980572162232546441, −2.93938536413558492458171503796, −2.46298339502555576257014305650, −0.910964092225331213605487742194, 1.59773491175408125762888560877, 2.34056370714565786268422924015, 4.05882710843299519677801274931, 4.37752325744730168189889159652, 5.35581045931930934276185538308, 6.46614309106027449945951199036, 6.77128800085977500787844601137, 7.74056671671815302960075045739, 8.747487200653167488117348808883, 9.397362012261536600515227394685

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