Properties

Label 2-690-23.4-c1-0-2
Degree $2$
Conductor $690$
Sign $-0.906 - 0.421i$
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  + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.743 + 5.17i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (1.65 − 3.62i)11-s + (0.415 − 0.909i)12-s + (−0.717 + 4.99i)13-s + (−4.39 + 2.82i)14-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (−1.53 − 1.77i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.376 + 0.241i)5-s + (−0.267 − 0.308i)6-s + (0.281 + 1.95i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0450 + 0.313i)10-s + (0.499 − 1.09i)11-s + (0.119 − 0.262i)12-s + (−0.199 + 1.38i)13-s + (−1.17 + 0.754i)14-s + (−0.247 − 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (−0.373 − 0.430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.906 - 0.421i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.906 - 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.291205 + 1.31732i\)
\(L(\frac12)\) \(\approx\) \(0.291205 + 1.31732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (3.56 - 3.20i)T \)
good7 \( 1 + (-0.743 - 5.17i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-1.65 + 3.62i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.717 - 4.99i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.53 + 1.77i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-1.97 + 2.27i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-4.46 - 5.15i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (7.85 + 2.30i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (0.311 - 0.200i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-1.62 - 1.04i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (10.6 - 3.13i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 4.92T + 47T^{2} \)
53 \( 1 + (0.316 + 2.20i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.310 + 2.16i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-12.6 - 3.71i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (0.852 + 1.86i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-3.15 - 6.90i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-9.98 + 11.5i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.615 + 4.28i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (7.64 - 4.91i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-3.93 + 1.15i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-6.57 - 4.22i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.24110551776989352684217874991, −9.637173882009516558287502074989, −9.071161630617120922264431854127, −8.420441152560951761420048894434, −7.01538487661445780794450670727, −6.23897939277856820042721726577, −5.55570610046635123147572148306, −4.80954681088984913953428233651, −3.37037493405390818042872390410, −2.03439640801910247393059198527, 0.70786598194517645321724438474, 1.89226505718905379227315450886, 3.67650124817346381262125020267, 4.44850849300490859374582490172, 5.35009901303946037518309490227, 6.53904016668800969421578815411, 7.38681471064394445665690475980, 8.266553562794962488415244928606, 9.797763940853512249465564020667, 10.22724506145363852031863012785

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