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Properties

Label 2-1690-13.10-c1-0-22
Degree $2$
Conductor $1690$
Sign $0.865 + 0.500i$
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  + (−0.866 + 0.5i)2-s + (−0.494 − 0.856i)3-s + (0.499 − 0.866i)4-s − i·5-s + (0.856 + 0.494i)6-s + (3.00 + 1.73i)7-s + 0.999i·8-s + (1.01 − 1.75i)9-s + (0.5 + 0.866i)10-s + (5.23 − 3.02i)11-s − 0.988·12-s − 3.47·14-s + (−0.856 + 0.494i)15-s + (−0.5 − 0.866i)16-s + (1.02 − 1.77i)17-s + 2.02i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.285 − 0.494i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (0.349 + 0.201i)6-s + (1.13 + 0.656i)7-s + 0.353i·8-s + (0.337 − 0.583i)9-s + (0.158 + 0.273i)10-s + (1.57 − 0.910i)11-s − 0.285·12-s − 0.928·14-s + (−0.221 + 0.127i)15-s + (−0.125 − 0.216i)16-s + (0.249 − 0.431i)17-s + 0.476i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.865 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.598430421\)
\(L(\frac12)\) \(\approx\) \(1.598430421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + iT \)
13 \( 1 \)
good3 \( 1 + (0.494 + 0.856i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.00 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.23 + 3.02i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.02 + 1.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 1.55i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.98 - 6.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.36 - 5.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.96iT - 31T^{2} \)
37 \( 1 + (-2.33 + 1.34i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.80 - 1.61i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.01 - 1.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.03iT - 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + (-5.54 - 3.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.52 - 11.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.35 + 4.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.105 - 0.0609i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.138iT - 73T^{2} \)
79 \( 1 + 6.71T + 79T^{2} \)
83 \( 1 - 3.03iT - 83T^{2} \)
89 \( 1 + (-2.93 + 1.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.04 + 2.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.310585575508844719215962855839, −8.491316777845958568305030071800, −7.74057002859922430815840465871, −6.98041882788944482181339301721, −6.04658207953492322878748343509, −5.50944142495398322320614702085, −4.43131889821845408247526971588, −3.23949123104410375274722396078, −1.53785075250510088499890845716, −1.07157682320014168123459659723, 1.18117926598901915731069210005, 2.09776025331874173494043955686, 3.54091768690800259879504019528, 4.47004990580351583336140862175, 4.96695604197789616302805811106, 6.52552628081340496691100803067, 7.05795485809869329647684424931, 7.908148089691075493577494374122, 8.660902295199258301467241113269, 9.607283683887913733612181530757

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