Properties

Label 2-690-23.13-c1-0-0
Degree $2$
Conductor $690$
Sign $-0.472 - 0.881i$
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 + (−1.69 − 0.496i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (−3.55 + 4.09i)11-s + (−0.654 + 0.755i)12-s + (−6.78 + 1.99i)13-s + (0.731 + 1.60i)14-s + (0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.457 + 3.18i)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 + (−0.638 − 0.187i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.303 + 0.0890i)10-s + (−1.07 + 1.23i)11-s + (−0.189 + 0.218i)12-s + (−1.88 + 0.552i)13-s + (0.195 + 0.428i)14-s + (0.217 − 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.110 + 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.472 - 0.881i$
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: \((2,\ 690,\ (\ :1/2),\ -0.472 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242497 + 0.405244i\)
\(L(\frac12)\) \(\approx\) \(0.242497 + 0.405244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.70 - 0.928i)T \)
good7 \( 1 + (1.69 + 0.496i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (3.55 - 4.09i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (6.78 - 1.99i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.457 - 3.18i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.185 + 1.29i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.0895 + 0.622i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-9.00 + 5.78i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-1.54 - 3.39i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (4.92 - 10.7i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.53 + 0.987i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 2.28T + 47T^{2} \)
53 \( 1 + (2.39 + 0.702i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (4.96 - 1.45i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (2.15 - 1.38i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (4.74 + 5.47i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-8.17 - 9.42i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.615 - 4.27i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-7.29 + 2.14i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (5.72 + 12.5i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-7.86 - 5.05i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (3.20 - 7.01i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.18173585988081281611173113718, −9.969161896276207131342999183860, −9.407275157739130512316708697359, −8.131916350694016721140246756758, −7.60070839432349222946568967730, −6.50105970190262357658430605257, −4.94346474880050010213957258204, −4.31044049521697908283340264847, −2.83608170910425985698596087873, −2.00974359862943218341764639461, 0.24774068902082085821975988104, 2.43326153051928284569756480351, 3.17835573179019871578936479217, 4.96540331707795975830401732136, 5.84406531797146259153892474748, 6.80725527652635969095874571615, 7.65986842172816559721150174066, 8.269997545110809750487640851362, 9.332568573964446787706355300715, 10.06323743556749504571236356147

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