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Properties

Label 2-1690-13.10-c1-0-15
Degree $2$
Conductor $1690$
Sign $-0.711 - 0.702i$
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 + (1.36 + 2.36i)3-s + (0.499 − 0.866i)4-s + i·5-s + (−2.36 − 1.36i)6-s + (2.59 + 1.5i)7-s + 0.999i·8-s + (−2.23 + 3.86i)9-s + (−0.5 − 0.866i)10-s + (2.59 − 1.5i)11-s + 2.73·12-s − 3·14-s + (−2.36 + 1.36i)15-s + (−0.5 − 0.866i)16-s + (1.09 − 1.90i)17-s − 4.46i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.788 + 1.36i)3-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−0.965 − 0.557i)6-s + (0.981 + 0.566i)7-s + 0.353i·8-s + (−0.744 + 1.28i)9-s + (−0.158 − 0.273i)10-s + (0.783 − 0.452i)11-s + 0.788·12-s − 0.801·14-s + (−0.610 + 0.352i)15-s + (−0.125 − 0.216i)16-s + (0.266 − 0.461i)17-s − 1.05i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 1690,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.038372473\)
\(L(\frac12)\) \(\approx\) \(2.038372473\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 2.36i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.09 + 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.59 - 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.26 + 2.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.73 - 8.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (9.69 - 5.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9 + 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (9 + 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 + (-14.8 + 8.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.0 + 7.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.483758235769160181691017391172, −8.895017545788119064533364676861, −8.308595760988381342878340800790, −7.56333810308468374998184528403, −6.49568993019805168759082254544, −5.40211725810051437105435401527, −4.80015918454119117945836965079, −3.62759743551168212523366692970, −2.89124411588070991735693284193, −1.55279970216025816246411512947, 0.998263171850622820431014178808, 1.56516053968533181871759089463, 2.58569071672347110502669688552, 3.76209546831043714624299528741, 4.78763656256207063173675766592, 6.11678839380510351796648637823, 7.05560343314242447906819851949, 7.70187795511722929174159118592, 8.060556288145542393428826744749, 8.993787123765161922325620569704

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