Properties

Label 2-690-23.4-c1-0-8
Degree $2$
Conductor $690$
Sign $-0.161 - 0.986i$
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 + (0.0903 + 0.628i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (−1.92 + 4.22i)11-s + (−0.415 + 0.909i)12-s + (−0.580 + 4.03i)13-s + (−0.533 + 0.343i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (−0.821 − 0.947i)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 + (0.0341 + 0.237i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0450 + 0.313i)10-s + (−0.581 + 1.27i)11-s + (−0.119 + 0.262i)12-s + (−0.160 + 1.11i)13-s + (−0.142 + 0.0917i)14-s + (0.247 + 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (−0.199 − 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.161 - 0.986i$
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: \((2,\ 690,\ (\ :1/2),\ -0.161 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31680 + 1.54956i\)
\(L(\frac12)\) \(\approx\) \(1.31680 + 1.54956i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.65 - 1.14i)T \)
good7 \( 1 + (-0.0903 - 0.628i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (1.92 - 4.22i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.580 - 4.03i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (0.821 + 0.947i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-0.992 + 1.14i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-3.08 - 3.55i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-4.34 - 1.27i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-8.68 + 5.58i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (7.73 + 4.96i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (7.97 - 2.34i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 5.16T + 47T^{2} \)
53 \( 1 + (-0.247 - 1.72i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.112 + 0.784i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (9.73 + 2.85i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (0.0920 + 0.201i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-2.51 - 5.50i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-10.6 + 12.3i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.88 + 13.1i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-0.774 + 0.497i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-2.72 + 0.799i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-5.73 - 3.68i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.54814619679676624536383477554, −9.557439214060182763004678174510, −9.033329000454300216822894078303, −7.953517600950078263438394373102, −7.06970715330839150731484803277, −6.55854291444246431052671859262, −5.16900617029176484969795923806, −4.47088992388582246139393995163, −3.05408234905318957663372524754, −1.95598324986966265086174153376, 0.969615477930258690204059001873, 2.64651499140812107536862473817, 3.33849061820770487243269658967, 4.64340555324352815911982873925, 5.50991623306799657152365812617, 6.51573186975083679537664647492, 8.054077617220720607219077349547, 8.410223089884747868196364652086, 9.593740163164006132203038017073, 10.24963042737912263955291401913

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