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Properties

Label 2-1710-95.18-c1-0-9
Degree $2$
Conductor $1710$
Sign $-0.563 - 0.826i$
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 + (2.23 + 0.0685i)5-s + (−2.16 + 2.16i)7-s + (0.707 − 0.707i)8-s + (−1.53 − 1.62i)10-s + 5.68·11-s + (−3.92 + 3.92i)13-s + 3.06·14-s − 1.00·16-s + (−4.99 + 4.99i)17-s + (−4.01 + 1.68i)19-s + (−0.0685 + 2.23i)20-s + (−4.01 − 4.01i)22-s + (−4.30 − 4.30i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.999 + 0.0306i)5-s + (−0.818 + 0.818i)7-s + (0.250 − 0.250i)8-s + (−0.484 − 0.515i)10-s + 1.71·11-s + (−1.08 + 1.08i)13-s + 0.818·14-s − 0.250·16-s + (−1.21 + 1.21i)17-s + (−0.922 + 0.387i)19-s + (−0.0153 + 0.499i)20-s + (−0.856 − 0.856i)22-s + (−0.897 − 0.897i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.563 - 0.826i$
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: \((2,\ 1710,\ (\ :1/2),\ -0.563 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6619346759\)
\(L(\frac12)\) \(\approx\) \(0.6619346759\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0685i)T \)
19 \( 1 + (4.01 - 1.68i)T \)
good7 \( 1 + (2.16 - 2.16i)T - 7iT^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
13 \( 1 + (3.92 - 3.92i)T - 13iT^{2} \)
17 \( 1 + (4.99 - 4.99i)T - 17iT^{2} \)
23 \( 1 + (4.30 + 4.30i)T + 23iT^{2} \)
29 \( 1 + 4.04T + 29T^{2} \)
31 \( 1 - 4.04iT - 31T^{2} \)
37 \( 1 + (6.19 + 6.19i)T + 37iT^{2} \)
41 \( 1 + 6.38iT - 41T^{2} \)
43 \( 1 + (4.15 + 4.15i)T + 43iT^{2} \)
47 \( 1 + (3.24 - 3.24i)T - 47iT^{2} \)
53 \( 1 + (-5.40 + 5.40i)T - 53iT^{2} \)
59 \( 1 + 2.39T + 59T^{2} \)
61 \( 1 + 2.33T + 61T^{2} \)
67 \( 1 + (-6.12 - 6.12i)T + 67iT^{2} \)
71 \( 1 + 6.51iT - 71T^{2} \)
73 \( 1 + (2.07 + 2.07i)T + 73iT^{2} \)
79 \( 1 + 4.23T + 79T^{2} \)
83 \( 1 + (-5.30 - 5.30i)T + 83iT^{2} \)
89 \( 1 + 2.50T + 89T^{2} \)
97 \( 1 + (6.22 + 6.22i)T + 97iT^{2} \)
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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

−9.478534991976331711243906604644, −8.960644351938498396343887145875, −8.555380662633433274837760569013, −6.87222661056447747055405912134, −6.58977383997270891192264547318, −5.78720273583004805932754773345, −4.43155792928029958910562948287, −3.65695582902816849511956659073, −2.14022753532180557447813245632, −1.92013379303700057424232316655, 0.27089840288741839166509887570, 1.67517762396396035595493943511, 2.88937260005759082976435423461, 4.13947283530641611695705117636, 5.08120677072671812774709027564, 6.10766254904068895939149307386, 6.74312631899795348245945064336, 7.20930171533195511405048349972, 8.353706295363059813070875950352, 9.328050032194632374059454959129

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