Properties

Label 2-1710-5.4-c1-0-43
Degree $2$
Conductor $1710$
Sign $-0.774 + 0.632i$
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  + i·2-s − 4-s + (−1.41 − 1.73i)5-s − 3.86i·7-s i·8-s + (1.73 − 1.41i)10-s + 1.03·11-s − 3.86i·13-s + 3.86·14-s + 16-s − 0.535i·17-s + 19-s + (1.41 + 1.73i)20-s + 1.03i·22-s + 5.46i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.632 − 0.774i)5-s − 1.46i·7-s − 0.353i·8-s + (0.547 − 0.447i)10-s + 0.312·11-s − 1.07i·13-s + 1.03·14-s + 0.250·16-s − 0.129i·17-s + 0.229·19-s + (0.316 + 0.387i)20-s + 0.220i·22-s + 1.13i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6018483060\)
\(L(\frac12)\) \(\approx\) \(0.6018483060\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (1.41 + 1.73i)T \)
19 \( 1 - T \)
good7 \( 1 + 3.86iT - 7T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
13 \( 1 + 3.86iT - 13T^{2} \)
17 \( 1 + 0.535iT - 17T^{2} \)
23 \( 1 - 5.46iT - 23T^{2} \)
29 \( 1 + 4.62T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 1.79iT - 37T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 + 6.69iT - 43T^{2} \)
47 \( 1 + 2.53iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6.96T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 - 4.14T + 71T^{2} \)
73 \( 1 + 3.58iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 16.3iT - 83T^{2} \)
89 \( 1 - 3.86T + 89T^{2} \)
97 \( 1 + 2.55iT - 97T^{2} \)
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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

−9.021930003219768839449386819763, −7.80899404230242067607155003195, −7.66458713206523764963557829465, −6.86140406319731267337723455113, −5.63779818243031115350259486873, −5.02225043320747262497681372054, −3.92811548719046353221651819870, −3.51775028545417087423008263130, −1.37230937991391700061551859162, −0.23763522483783411116558781253, 1.84942581842825368940003066091, 2.65514758537175743429139910168, 3.62491725318626617332294996424, 4.47978032656668908234152881390, 5.57552933271637516957081922934, 6.39553159754247584205102221644, 7.28791836891210189037956773298, 8.251985532480712659226586378321, 9.043434766974736097475560794901, 9.484222606964455400685124986522

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