Properties

Label 2-690-115.7-c1-0-0
Degree $2$
Conductor $690$
Sign $-0.913 - 0.406i$
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  + (−0.997 + 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (1.98 + 1.03i)5-s + (0.142 − 0.989i)6-s + (−0.653 + 1.19i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (−2.05 − 0.888i)10-s + (−1.60 + 1.38i)11-s + (−0.0713 + 0.997i)12-s + (−4.94 + 2.70i)13-s + (0.566 − 1.24i)14-s + (−1.43 + 1.71i)15-s + (0.959 − 0.281i)16-s + (−1.26 − 1.68i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.886 + 0.462i)5-s + (0.0580 − 0.404i)6-s + (−0.247 + 0.452i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.648 − 0.281i)10-s + (−0.482 + 0.418i)11-s + (−0.0205 + 0.287i)12-s + (−1.37 + 0.749i)13-s + (0.151 − 0.331i)14-s + (−0.369 + 0.443i)15-s + (0.239 − 0.0704i)16-s + (−0.306 − 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.913 - 0.406i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144548 + 0.681320i\)
\(L(\frac12)\) \(\approx\) \(0.144548 + 0.681320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 - 0.0713i)T \)
3 \( 1 + (0.212 - 0.977i)T \)
5 \( 1 + (-1.98 - 1.03i)T \)
23 \( 1 + (4.28 + 2.15i)T \)
good7 \( 1 + (0.653 - 1.19i)T + (-3.78 - 5.88i)T^{2} \)
11 \( 1 + (1.60 - 1.38i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (4.94 - 2.70i)T + (7.02 - 10.9i)T^{2} \)
17 \( 1 + (1.26 + 1.68i)T + (-4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.299 - 2.08i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-5.11 - 0.735i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.875 + 0.562i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-3.05 - 8.19i)T + (-27.9 + 24.2i)T^{2} \)
41 \( 1 + (3.31 + 7.25i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (6.00 + 1.30i)T + (39.1 + 17.8i)T^{2} \)
47 \( 1 + (1.39 + 1.39i)T + 47iT^{2} \)
53 \( 1 + (2.68 + 1.46i)T + (28.6 + 44.5i)T^{2} \)
59 \( 1 + (0.317 - 1.08i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-0.435 + 0.678i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-1.14 - 16.0i)T + (-66.3 + 9.53i)T^{2} \)
71 \( 1 + (7.09 - 8.18i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.494 - 0.369i)T + (20.5 + 70.0i)T^{2} \)
79 \( 1 + (13.0 + 3.84i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (1.39 - 0.520i)T + (62.7 - 54.3i)T^{2} \)
89 \( 1 + (-14.7 + 9.49i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-10.5 - 3.94i)T + (73.3 + 63.5i)T^{2} \)
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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

−10.30736338258288087754519741339, −10.08218996249561124476520724339, −9.339375447805332171580879146166, −8.474043174697759899409085132280, −7.27437628022721341351995953970, −6.51680242636327819080643872899, −5.53904710671726254056882324191, −4.57389116021497138004022510406, −2.89198349835720103099075672760, −2.05441826084251451193226278690, 0.43111146209327212977559827696, 1.92962849097815613111879860633, 2.98846487460558581180279077916, 4.78340617431344179317987548390, 5.77405223395717800746369479082, 6.62243411553746017999112485609, 7.61471336733893249404244205165, 8.293656104638462488050320437033, 9.336655751931082835483434910613, 10.06406949380732608320087267821

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