Properties

Label 2-690-23.13-c1-0-1
Degree $2$
Conductor $690$
Sign $-0.936 - 0.350i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (−0.142 − 0.989i)6-s + (−2.28 − 0.672i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (−2.98 + 3.44i)11-s + (0.654 − 0.755i)12-s + (−1.03 + 0.303i)13-s + (−0.991 − 2.17i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.0307 + 0.213i)17-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.185 − 0.406i)5-s + (−0.0580 − 0.404i)6-s + (−0.865 − 0.254i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (0.303 − 0.0890i)10-s + (−0.901 + 1.04i)11-s + (0.189 − 0.218i)12-s + (−0.286 + 0.0842i)13-s + (−0.264 − 0.580i)14-s + (−0.217 + 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.00746 + 0.0518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.936 - 0.350i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107881 + 0.596262i\)
\(L(\frac12)\) \(\approx\) \(0.107881 + 0.596262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (0.841 + 0.540i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (2.92 - 3.80i)T \)
good7 \( 1 + (2.28 + 0.672i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (2.98 - 3.44i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (1.03 - 0.303i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.0307 - 0.213i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.801 - 5.57i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-0.113 - 0.786i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (4.68 - 3.00i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (2.14 + 4.68i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (0.401 - 0.879i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-5.15 - 3.31i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 + (3.38 + 0.994i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-5.22 + 1.53i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (9.10 - 5.85i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (9.12 + 10.5i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (2.21 + 2.56i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.94 + 13.4i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (0.0557 - 0.0163i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-6.91 - 15.1i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-1.58 - 1.02i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-3.73 + 8.18i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.75658892170864119453902724949, −9.994864796100688018388782239034, −9.182805778669154135013828725672, −7.85995574154252911852034255548, −7.34651440233109469711881490995, −6.30503469276045190495611902890, −5.55143136941023594345931687001, −4.64078946348568784226783758547, −3.50459669223941853645908926125, −1.96420227956606014303942630178, 0.27029809948712397446854761817, 2.49147871865157318592861181780, 3.28532305348900066334745572077, 4.52430716223257622253732492741, 5.59913993006643856487797522006, 6.21395723542619278964233791049, 7.22642938940382166776108617121, 8.588083598565550153141517068163, 9.489266658850361619011099053864, 10.32024360297886287902521302241

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