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Properties

Label 2-138-23.22-c4-0-5
Degree $2$
Conductor $138$
Sign $0.932 + 0.360i$
Analytic cond. $14.2650$
Root an. cond. $3.77691$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s − 5.19·3-s + 8.00·4-s − 26.7i·5-s + 14.6·6-s + 66.7i·7-s − 22.6·8-s + 27·9-s + 75.5i·10-s + 8.81i·11-s − 41.5·12-s − 136.·13-s − 188. i·14-s + 138. i·15-s + 64.0·16-s − 184. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.500·4-s − 1.06i·5-s + 0.408·6-s + 1.36i·7-s − 0.353·8-s + 0.333·9-s + 0.755i·10-s + 0.0728i·11-s − 0.288·12-s − 0.810·13-s − 0.962i·14-s + 0.616i·15-s + 0.250·16-s − 0.636i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(14.2650\)
Root analytic conductor: \(3.77691\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :2),\ 0.932 + 0.360i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.952368 - 0.177430i\)
\(L(\frac12)\) \(\approx\) \(0.952368 - 0.177430i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
3 \( 1 + 5.19T \)
23 \( 1 + (-493. - 190. i)T \)
good5 \( 1 + 26.7iT - 625T^{2} \)
7 \( 1 - 66.7iT - 2.40e3T^{2} \)
11 \( 1 - 8.81iT - 1.46e4T^{2} \)
13 \( 1 + 136.T + 2.85e4T^{2} \)
17 \( 1 + 184. iT - 8.35e4T^{2} \)
19 \( 1 - 33.2iT - 1.30e5T^{2} \)
29 \( 1 - 1.22e3T + 7.07e5T^{2} \)
31 \( 1 - 1.22e3T + 9.23e5T^{2} \)
37 \( 1 + 2.05e3iT - 1.87e6T^{2} \)
41 \( 1 - 884.T + 2.82e6T^{2} \)
43 \( 1 + 1.57e3iT - 3.41e6T^{2} \)
47 \( 1 - 564.T + 4.87e6T^{2} \)
53 \( 1 + 3.03e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.66e3T + 1.21e7T^{2} \)
61 \( 1 - 3.37e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.98e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.48e3T + 2.54e7T^{2} \)
73 \( 1 - 406.T + 2.83e7T^{2} \)
79 \( 1 + 103. iT - 3.89e7T^{2} \)
83 \( 1 + 1.01e4iT - 4.74e7T^{2} \)
89 \( 1 + 7.41e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.61e4iT - 8.85e7T^{2} \)
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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

−12.15127288312996173334948208629, −11.65282883331495104461406403113, −10.19602666750726044577206257442, −9.175964738208689963558093582448, −8.496081770940065265203499411405, −7.10302360030928280544260804672, −5.69887034803397258222255226269, −4.82003187819889042415140951801, −2.47681518747390284069564045058, −0.77484949072739970587906029318, 0.904654454708554381950284696790, 2.92139308390534978537524561265, 4.57466389002238150474633815184, 6.45922559938739017027290151177, 7.03902603909697835457241321274, 8.134645974684223118947928110191, 9.844991367377491146887742324697, 10.48926638511571413571214004646, 11.14378274554891895750206038137, 12.30796162910209212375523531892

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