Properties

Label 2-690-115.7-c1-0-10
Degree $2$
Conductor $690$
Sign $0.999 - 0.0131i$
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 + (−1.79 + 1.33i)5-s + (−0.142 + 0.989i)6-s + (2.35 − 4.31i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (−1.69 + 1.45i)10-s + (0.567 − 0.491i)11-s + (−0.0713 + 0.997i)12-s + (5.56 − 3.03i)13-s + (2.04 − 4.47i)14-s + (−0.920 − 2.03i)15-s + (0.959 − 0.281i)16-s + (0.838 + 1.12i)17-s + ⋯
L(s)  = 1  + (0.705 − 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (−0.802 + 0.596i)5-s + (−0.0580 + 0.404i)6-s + (0.891 − 1.63i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.536 + 0.460i)10-s + (0.171 − 0.148i)11-s + (−0.0205 + 0.287i)12-s + (1.54 − 0.842i)13-s + (0.546 − 1.19i)14-s + (−0.237 − 0.526i)15-s + (0.239 − 0.0704i)16-s + (0.203 + 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.999 - 0.0131i$
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: \((2,\ 690,\ (\ :1/2),\ 0.999 - 0.0131i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25277 + 0.0148432i\)
\(L(\frac12)\) \(\approx\) \(2.25277 + 0.0148432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 - 1.33i)T \)
23 \( 1 + (-1.94 - 4.38i)T \)
good7 \( 1 + (-2.35 + 4.31i)T + (-3.78 - 5.88i)T^{2} \)
11 \( 1 + (-0.567 + 0.491i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (-5.56 + 3.03i)T + (7.02 - 10.9i)T^{2} \)
17 \( 1 + (-0.838 - 1.12i)T + (-4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.534 - 3.71i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (6.83 + 0.983i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-6.38 - 4.10i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-1.34 - 3.61i)T + (-27.9 + 24.2i)T^{2} \)
41 \( 1 + (0.469 + 1.02i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (2.55 + 0.555i)T + (39.1 + 17.8i)T^{2} \)
47 \( 1 + (8.63 + 8.63i)T + 47iT^{2} \)
53 \( 1 + (-9.22 - 5.03i)T + (28.6 + 44.5i)T^{2} \)
59 \( 1 + (-1.73 + 5.89i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (0.0184 - 0.0286i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-0.540 - 7.55i)T + (-66.3 + 9.53i)T^{2} \)
71 \( 1 + (-2.93 + 3.38i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (7.74 + 5.79i)T + (20.5 + 70.0i)T^{2} \)
79 \( 1 + (11.4 + 3.37i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (0.379 - 0.141i)T + (62.7 - 54.3i)T^{2} \)
89 \( 1 + (-10.9 + 7.04i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (2.96 + 1.10i)T + (73.3 + 63.5i)T^{2} \)
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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

−10.54278909860715971616768946447, −10.18253442479298870615215486258, −8.438914045812329524899890044759, −7.81110693483511451758098271118, −6.94336240001470793693671867965, −5.88076506759485780357211281668, −4.77075105217620910292639965582, −3.69302327443153286289313664145, −3.51687389247665453216123060164, −1.23276333878295203555517071362, 1.46415483804795729809045305682, 2.70131520008284755320105610244, 4.12833137674263335330175696793, 5.02969844474918958541329628479, 5.87266935362207642032629715046, 6.79288318122524522202392567097, 7.947190664671498939554809237260, 8.603691459998741363984172608401, 9.242762290791127314091355525980, 11.16472870192027749635433606082

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