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Properties

Label 2-1690-13.4-c1-0-31
Degree $2$
Conductor $1690$
Sign $0.702 + 0.711i$
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 + (0.499 + 0.866i)4-s + i·5-s + (2.59 − 1.5i)7-s − 0.999i·8-s + (1.5 + 2.59i)9-s + (0.5 − 0.866i)10-s + (−2.59 − 1.5i)11-s − 3·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s − 3i·18-s + (6.06 − 3.5i)19-s + (−0.866 + 0.499i)20-s + (1.5 + 2.59i)22-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (0.981 − 0.566i)7-s − 0.353i·8-s + (0.5 + 0.866i)9-s + (0.158 − 0.273i)10-s + (−0.783 − 0.452i)11-s − 0.801·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s − 0.707i·18-s + (1.39 − 0.802i)19-s + (−0.193 + 0.111i)20-s + (0.319 + 0.553i)22-s + (−0.417 + 0.722i)23-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.398021753\)
\(L(\frac12)\) \(\approx\) \(1.398021753\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T^{2} \)
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 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.06 + 3.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.73 - i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3.46 + 2i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.8 + 8i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.512293116977763338237257208312, −8.245618111007755084469587407719, −7.64375055124116547620785342712, −7.35897460575907700356817633667, −6.06920369703444240244717949588, −4.98158678675071788694267795212, −4.27977547609397809671781115823, −2.93939525712789090217671066086, −2.13264137224639176780281279948, −0.78458933043696564806228940879, 1.12948816662578062265994010458, 2.07823451131039129602787859291, 3.50645858452693664003575763873, 4.76964472002307454901399210495, 5.32283139925858486269031444165, 6.32572937015522899292399836744, 7.18165083240653142150867848637, 8.028681794452184114661469580430, 8.585970772178402320096110883582, 9.287882700381742119154597940343

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