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Properties

Label 2-138-23.22-c4-0-6
Degree $2$
Conductor $138$
Sign $0.725 - 0.688i$
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 + 19.9i·5-s − 14.6·6-s − 58.9i·7-s + 22.6·8-s + 27·9-s + 56.4i·10-s + 168. i·11-s − 41.5·12-s + 288.·13-s − 166. i·14-s − 103. i·15-s + 64.0·16-s + 311. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.798i·5-s − 0.408·6-s − 1.20i·7-s + 0.353·8-s + 0.333·9-s + 0.564i·10-s + 1.39i·11-s − 0.288·12-s + 1.70·13-s − 0.851i·14-s − 0.461i·15-s + 0.250·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.725 - 0.688i$
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: \((2,\ 138,\ (\ :2),\ 0.725 - 0.688i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.23012 + 0.889732i\)
\(L(\frac12)\) \(\approx\) \(2.23012 + 0.889732i\)
\(L(3)\) 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 \( 1 + (383. - 364. i)T \)
good5 \( 1 - 19.9iT - 625T^{2} \)
7 \( 1 + 58.9iT - 2.40e3T^{2} \)
11 \( 1 - 168. iT - 1.46e4T^{2} \)
13 \( 1 - 288.T + 2.85e4T^{2} \)
17 \( 1 - 311. iT - 8.35e4T^{2} \)
19 \( 1 - 118. iT - 1.30e5T^{2} \)
29 \( 1 - 991.T + 7.07e5T^{2} \)
31 \( 1 - 827.T + 9.23e5T^{2} \)
37 \( 1 - 80.7iT - 1.87e6T^{2} \)
41 \( 1 - 187.T + 2.82e6T^{2} \)
43 \( 1 - 331. iT - 3.41e6T^{2} \)
47 \( 1 + 1.89e3T + 4.87e6T^{2} \)
53 \( 1 + 4.19e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.96e3T + 1.21e7T^{2} \)
61 \( 1 + 5.56e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.83e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.91e3T + 2.54e7T^{2} \)
73 \( 1 + 6.60e3T + 2.83e7T^{2} \)
79 \( 1 + 1.13e4iT - 3.89e7T^{2} \)
83 \( 1 - 3.57e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.64e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.50e3iT - 8.85e7T^{2} \)
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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

−12.70749839934523854729228983836, −11.56257743524282506251009591122, −10.61517230199706229461997034734, −10.08526642983134438314090413411, −8.025540372380865120375579582756, −6.86272954025787954172798826223, −6.17404835844895887399680991207, −4.51870727710932584281063573038, −3.55781454011175491406763053560, −1.49763274861329333310038374561, 0.963779725728799674125977155099, 2.95974864191187110722597479501, 4.59021496278937717358215991100, 5.71539552784577248064941730206, 6.35853853434091387505497990989, 8.308054466014742429221851812419, 9.007682593493621209139396170933, 10.68942151760235742706757663930, 11.61543443674230444885347864985, 12.27293279607593286418713827260

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