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Properties

Label 2-138-23.22-c4-0-12
Degree $2$
Conductor $138$
Sign $-0.225 + 0.974i$
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 − 3.47i·5-s − 14.6·6-s − 45.9i·7-s + 22.6·8-s + 27·9-s − 9.82i·10-s + 39.1i·11-s − 41.5·12-s − 239.·13-s − 129. i·14-s + 18.0i·15-s + 64.0·16-s − 452. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.500·4-s − 0.138i·5-s − 0.408·6-s − 0.937i·7-s + 0.353·8-s + 0.333·9-s − 0.0982i·10-s + 0.323i·11-s − 0.288·12-s − 1.41·13-s − 0.663i·14-s + 0.0802i·15-s + 0.250·16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.225 + 0.974i$
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: \((2,\ 138,\ (\ :2),\ -0.225 + 0.974i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.05611 - 1.32829i\)
\(L(\frac12)\) \(\approx\) \(1.05611 - 1.32829i\)
\(L(3)\) 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 \( 1 + (-119. + 515. i)T \)
good5 \( 1 + 3.47iT - 625T^{2} \)
7 \( 1 + 45.9iT - 2.40e3T^{2} \)
11 \( 1 - 39.1iT - 1.46e4T^{2} \)
13 \( 1 + 239.T + 2.85e4T^{2} \)
17 \( 1 + 452. iT - 8.35e4T^{2} \)
19 \( 1 + 497. iT - 1.30e5T^{2} \)
29 \( 1 + 11.7T + 7.07e5T^{2} \)
31 \( 1 + 845.T + 9.23e5T^{2} \)
37 \( 1 + 1.37e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.98e3T + 2.82e6T^{2} \)
43 \( 1 - 3.43e3iT - 3.41e6T^{2} \)
47 \( 1 - 3.52e3T + 4.87e6T^{2} \)
53 \( 1 - 1.48e3iT - 7.89e6T^{2} \)
59 \( 1 - 41.9T + 1.21e7T^{2} \)
61 \( 1 + 1.56e3iT - 1.38e7T^{2} \)
67 \( 1 - 7.38e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.47e3T + 2.54e7T^{2} \)
73 \( 1 - 6.55e3T + 2.83e7T^{2} \)
79 \( 1 + 2.77e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.08e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.41e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.19e4iT - 8.85e7T^{2} \)
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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

−12.28554228674138419350746819766, −11.30371693676332509164360885518, −10.38724442722198447110417515366, −9.256999652261188738174891894681, −7.33351960947321823859806259116, −6.88636202597044468829394450085, −5.13346569466153142632846668771, −4.49041494518290109762677992362, −2.67354479248026111582792100149, −0.56922183975629538996676461930, 1.93284183572454206921938894349, 3.57700603004726414772871129368, 5.14637382645269227949519193814, 5.91780626947473779015987432947, 7.14815312652812412422598892672, 8.483111522538601188311019386280, 9.948939605645456008337779314750, 10.88742220471532286285646162872, 12.23599890182477002903557553301, 12.33524386522281005674264528952

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