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Properties

Label 2-1690-13.9-c1-0-19
Degree $2$
Conductor $1690$
Sign $0.979 + 0.202i$
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 + (0.300 − 0.519i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.300 − 0.519i)6-s + (0.719 + 1.24i)7-s − 0.999·8-s + (1.31 + 2.28i)9-s + (−0.5 + 0.866i)10-s + (1.38 − 2.40i)11-s − 0.600·12-s + 1.43·14-s + (−0.300 + 0.519i)15-s + (−0.5 + 0.866i)16-s + (2.25 + 3.90i)17-s + 2.63·18-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.173 − 0.300i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.122 − 0.212i)6-s + (0.272 + 0.471i)7-s − 0.353·8-s + (0.439 + 0.762i)9-s + (−0.158 + 0.273i)10-s + (0.417 − 0.723i)11-s − 0.173·12-s + 0.384·14-s + (−0.0774 + 0.134i)15-s + (−0.125 + 0.216i)16-s + (0.546 + 0.945i)17-s + 0.622·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.979 + 0.202i$
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: \((2,\ 1690,\ (\ :1/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.072660547\)
\(L(\frac12)\) \(\approx\) \(2.072660547\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.300 + 0.519i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.719 - 1.24i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.38 + 2.40i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.25 - 3.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.16 - 3.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.21 - 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.93 - 6.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + (0.287 - 0.498i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.11 + 3.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.30 + 2.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + (1.73 + 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.25 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.66 + 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.66T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (-2.97 + 5.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.793 + 1.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.265358522828720705907003672386, −8.559000955006958860780981150500, −7.79619221831395712978427174995, −7.05319655463482932525186459412, −5.73488235111947815500970011104, −5.35959195368105641922147778855, −4.00908676533977475398307962296, −3.48073408357222679634859524547, −2.16774357755972990975538317326, −1.28083133988626491157738094987, 0.790651598379130612543059763347, 2.57790842416654924066127991283, 3.81518143775374627859167937129, 4.28004409671120091430707942521, 5.16183668003138475455691396151, 6.23265616471749060089079510415, 7.20991564950278603968095332949, 7.44174090785046707686405831314, 8.538444374738571832115314378910, 9.396162589099214743648448749602

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