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Properties

Label 2-1690-13.9-c1-0-32
Degree $2$
Conductor $1690$
Sign $0.872 + 0.488i$
Analytic cond. $13.4947$
Root an. cond. $3.67351$
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  + (−0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 − 0.866i)4-s + 5-s + (0.999 + 1.73i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (1.5 − 2.59i)11-s − 1.99·12-s + 0.999·14-s + (1 − 1.73i)15-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.408 + 0.707i)6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.452 − 0.783i)11-s − 0.577·12-s + 0.267·14-s + (0.258 − 0.447i)15-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.993734486\)
\(L(\frac12)\) \(\approx\) \(1.993734486\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8 - 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.979784804722801552281861436416, −8.384546510584759237783629165674, −7.66906257601691770340816374440, −7.08570264490365240560929461565, −6.05137942896443940043817775605, −5.73185923010427629204586453964, −4.22351134240283034235869068133, −3.19775499798245185753824823934, −1.88706310074229075326406511398, −0.994087117750099534792808151022, 1.19983872734554340936730499331, 2.72750756204938116395956499403, 3.12175175297802783136867500716, 4.46478699465104188207115764858, 4.86718630481854922782114714168, 6.18522220972904126864990974891, 7.17850721585389244482157080965, 8.057947557732271344993713585285, 9.176358000644681766782708401091, 9.379682598892864739232980373535

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