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Properties

Label 2-1690-13.4-c1-0-4
Degree $2$
Conductor $1690$
Sign $0.455 - 0.890i$
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 + (0.900 − 1.56i)3-s + (0.499 + 0.866i)4-s − i·5-s + (−1.56 + 0.900i)6-s + (−3.12 + 1.80i)7-s − 0.999i·8-s + (−0.123 − 0.213i)9-s + (−0.5 + 0.866i)10-s + (−3.84 − 2.22i)11-s + 1.80·12-s + 3.60·14-s + (−1.56 − 0.900i)15-s + (−0.5 + 0.866i)16-s + (−1.57 − 2.73i)17-s + 0.246i·18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.520 − 0.900i)3-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (−0.637 + 0.367i)6-s + (−1.17 + 0.681i)7-s − 0.353i·8-s + (−0.0411 − 0.0712i)9-s + (−0.158 + 0.273i)10-s + (−1.16 − 0.670i)11-s + 0.520·12-s + 0.963·14-s + (−0.402 − 0.232i)15-s + (−0.125 + 0.216i)16-s + (−0.383 − 0.663i)17-s + 0.0582i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.455 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4391198116\)
\(L(\frac12)\) \(\approx\) \(0.4391198116\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.900 + 1.56i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.12 - 1.80i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.84 + 2.22i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.57 + 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.309 + 0.178i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.44 + 2.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.82iT - 31T^{2} \)
37 \( 1 + (-6.47 - 3.74i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.76 - 1.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.00 - 8.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + (10.3 - 5.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.18 - 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.1 - 6.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.0 - 6.40i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.96iT - 73T^{2} \)
79 \( 1 + 5.87T + 79T^{2} \)
83 \( 1 + 4.67iT - 83T^{2} \)
89 \( 1 + (6.94 + 4.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.42 - 1.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.503927503481084351982090220838, −8.657601115340691137185412281386, −7.899762868368526617837320933664, −7.47594185951702453086361507401, −6.32418913010336501284726657845, −5.70283462134202071880392640846, −4.38376157684301772839561772371, −2.82308333612339284132080406533, −2.69851383501713917916862144868, −1.23503241463210490168260247795, 0.19535850316481364256839121970, 2.24032389329821787360195764873, 3.24400490306298393552606703019, 4.07048435969195630258873595122, 5.03031888509638782438406123633, 6.28342386237916468131820135747, 6.81769522969648057473324400171, 7.69936648679621029697878653405, 8.492049426573526448290120938009, 9.389787309001768204675152065299

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