Properties

Label 2-138-69.68-c3-0-18
Degree $2$
Conductor $138$
Sign $0.0524 + 0.998i$
Analytic cond. $8.14226$
Root an. cond. $2.85346$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + (4.33 − 2.86i)3-s − 4·4-s + 13.9·5-s + (−5.72 − 8.67i)6-s + 0.843i·7-s + 8i·8-s + (10.5 − 24.8i)9-s − 27.9i·10-s + 54.5·11-s + (−17.3 + 11.4i)12-s − 40.5·13-s + 1.68·14-s + (60.6 − 40.0i)15-s + 16·16-s − 3.33·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.834 − 0.551i)3-s − 0.5·4-s + 1.25·5-s + (−0.389 − 0.590i)6-s + 0.0455i·7-s + 0.353i·8-s + (0.392 − 0.919i)9-s − 0.885i·10-s + 1.49·11-s + (−0.417 + 0.275i)12-s − 0.864·13-s + 0.0321·14-s + (1.04 − 0.689i)15-s + 0.250·16-s − 0.0476·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0524 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.0524 + 0.998i$
Analytic conductor: \(8.14226\)
Root analytic conductor: \(2.85346\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3/2),\ 0.0524 + 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.82137 - 1.72820i\)
\(L(\frac12)\) \(\approx\) \(1.82137 - 1.72820i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 + (-4.33 + 2.86i)T \)
23 \( 1 + (95.1 + 55.8i)T \)
good5 \( 1 - 13.9T + 125T^{2} \)
7 \( 1 - 0.843iT - 343T^{2} \)
11 \( 1 - 54.5T + 1.33e3T^{2} \)
13 \( 1 + 40.5T + 2.19e3T^{2} \)
17 \( 1 + 3.33T + 4.91e3T^{2} \)
19 \( 1 - 20.4iT - 6.85e3T^{2} \)
29 \( 1 - 35.1iT - 2.43e4T^{2} \)
31 \( 1 + 138.T + 2.97e4T^{2} \)
37 \( 1 + 55.4iT - 5.06e4T^{2} \)
41 \( 1 - 98.3iT - 6.89e4T^{2} \)
43 \( 1 - 457. iT - 7.95e4T^{2} \)
47 \( 1 + 256. iT - 1.03e5T^{2} \)
53 \( 1 - 158.T + 1.48e5T^{2} \)
59 \( 1 + 102. iT - 2.05e5T^{2} \)
61 \( 1 - 650. iT - 2.26e5T^{2} \)
67 \( 1 - 1.05e3iT - 3.00e5T^{2} \)
71 \( 1 - 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 - 56.9T + 3.89e5T^{2} \)
79 \( 1 + 266. iT - 4.93e5T^{2} \)
83 \( 1 + 321.T + 5.71e5T^{2} \)
89 \( 1 - 1.04e3T + 7.04e5T^{2} \)
97 \( 1 - 773. iT - 9.12e5T^{2} \)
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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

−12.55245595521946755512113851382, −11.68822428038467041786208343589, −10.08831464818940004567877059140, −9.461510598093305603382790865331, −8.579939367658663302781185451855, −7.05765281074115110674424255029, −5.88198121562305985879877325452, −4.07455477236119446084139325683, −2.51112370300735174398475053215, −1.43078116170214571983657135626, 1.99289051129356206012971912778, 3.82886312297803820672484186545, 5.18594509790081715708907240058, 6.42750033914413785224419967531, 7.63419462752220472828650869379, 9.081201275734081756404816870085, 9.461759509543311667132911044848, 10.47988187326097759982186861214, 12.14474281172366677336566441529, 13.48231447877081946587387376206

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