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Properties

Label 2-1710-57.56-c3-0-22
Degree $2$
Conductor $1710$
Sign $0.581 - 0.813i$
Analytic cond. $100.893$
Root an. cond. $10.0445$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 5i·5-s − 16.5·7-s − 8·8-s − 10i·10-s − 6.84i·11-s + 45.4i·13-s + 33.0·14-s + 16·16-s − 71.2i·17-s + (0.472 − 82.8i)19-s + 20i·20-s + 13.6i·22-s + 138. i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447i·5-s − 0.891·7-s − 0.353·8-s − 0.316i·10-s − 0.187i·11-s + 0.969i·13-s + 0.630·14-s + 0.250·16-s − 1.01i·17-s + (0.00570 − 0.999i)19-s + 0.223i·20-s + 0.132i·22-s + 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(100.893\)
Root analytic conductor: \(10.0445\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :3/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9527258949\)
\(L(\frac12)\) \(\approx\) \(0.9527258949\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 \)
5 \( 1 - 5iT \)
19 \( 1 + (-0.472 + 82.8i)T \)
good7 \( 1 + 16.5T + 343T^{2} \)
11 \( 1 + 6.84iT - 1.33e3T^{2} \)
13 \( 1 - 45.4iT - 2.19e3T^{2} \)
17 \( 1 + 71.2iT - 4.91e3T^{2} \)
23 \( 1 - 138. iT - 1.21e4T^{2} \)
29 \( 1 - 42.3T + 2.43e4T^{2} \)
31 \( 1 - 86.7iT - 2.97e4T^{2} \)
37 \( 1 + 410. iT - 5.06e4T^{2} \)
41 \( 1 + 412.T + 6.89e4T^{2} \)
43 \( 1 - 481.T + 7.95e4T^{2} \)
47 \( 1 + 268. iT - 1.03e5T^{2} \)
53 \( 1 - 72.1T + 1.48e5T^{2} \)
59 \( 1 - 125.T + 2.05e5T^{2} \)
61 \( 1 + 125.T + 2.26e5T^{2} \)
67 \( 1 + 82.5iT - 3.00e5T^{2} \)
71 \( 1 + 347.T + 3.57e5T^{2} \)
73 \( 1 - 796.T + 3.89e5T^{2} \)
79 \( 1 + 174. iT - 4.93e5T^{2} \)
83 \( 1 - 1.16e3iT - 5.71e5T^{2} \)
89 \( 1 + 874.T + 7.04e5T^{2} \)
97 \( 1 - 978. iT - 9.12e5T^{2} \)
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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

−9.400278287902504248604520959037, −8.446367422833572696985775791319, −7.19359103218428475674923662014, −7.04970325173063039419127991035, −6.08946530817299869162989430851, −5.12980667862090151293110068280, −3.85010891010734483237563892054, −2.97854323533024854486745656823, −2.05547791168346422427576248983, −0.65592408772851481624733192108, 0.40773894030635097256020693032, 1.50842319100817234106011148033, 2.73266597378543672556437228966, 3.63787323799272885312234333413, 4.73308293144352381756101729568, 5.93445503754925334564118908012, 6.35632274156228666949105702850, 7.43004886178429217150323452708, 8.267195490893406216220629024377, 8.685517002732559622258093287090

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