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Properties

Label 2-138-23.22-c4-0-1
Degree $2$
Conductor $138$
Sign $-0.928 - 0.370i$
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 + 34.5i·5-s − 14.6·6-s + 81.5i·7-s − 22.6·8-s + 27·9-s − 97.8i·10-s − 67.6i·11-s + 41.5·12-s − 316.·13-s − 230. i·14-s + 179. i·15-s + 64.0·16-s − 44.6i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.500·4-s + 1.38i·5-s − 0.408·6-s + 1.66i·7-s − 0.353·8-s + 0.333·9-s − 0.978i·10-s − 0.558i·11-s + 0.288·12-s − 1.87·13-s − 1.17i·14-s + 0.798i·15-s + 0.250·16-s − 0.154i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.928 - 0.370i$
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: \((2,\ 138,\ (\ :2),\ -0.928 - 0.370i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.178922 + 0.932119i\)
\(L(\frac12)\) \(\approx\) \(0.178922 + 0.932119i\)
\(L(3)\) 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 \( 1 + (-491. - 195. i)T \)
good5 \( 1 - 34.5iT - 625T^{2} \)
7 \( 1 - 81.5iT - 2.40e3T^{2} \)
11 \( 1 + 67.6iT - 1.46e4T^{2} \)
13 \( 1 + 316.T + 2.85e4T^{2} \)
17 \( 1 + 44.6iT - 8.35e4T^{2} \)
19 \( 1 + 468. iT - 1.30e5T^{2} \)
29 \( 1 + 228.T + 7.07e5T^{2} \)
31 \( 1 + 1.67e3T + 9.23e5T^{2} \)
37 \( 1 - 736. iT - 1.87e6T^{2} \)
41 \( 1 + 299.T + 2.82e6T^{2} \)
43 \( 1 - 2.67e3iT - 3.41e6T^{2} \)
47 \( 1 - 3.46e3T + 4.87e6T^{2} \)
53 \( 1 + 142. iT - 7.89e6T^{2} \)
59 \( 1 + 1.69e3T + 1.21e7T^{2} \)
61 \( 1 - 6.28e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.56e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.39e3T + 2.54e7T^{2} \)
73 \( 1 - 1.10e3T + 2.83e7T^{2} \)
79 \( 1 - 4.63e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.28e4iT - 4.74e7T^{2} \)
89 \( 1 + 3.84e3iT - 6.27e7T^{2} \)
97 \( 1 - 940. iT - 8.85e7T^{2} \)
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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

−12.78697772917092244187735075472, −11.69074917812950552899326766404, −10.85805763339246962677231438018, −9.565206787269727056733716585950, −8.976622008920295885261486806583, −7.60340542537812231319758250575, −6.76898623197286082588447663149, −5.34370624044536266960956032081, −2.93137078550774891333636637351, −2.40316586916300488642324918005, 0.42771448406921908060802707692, 1.81067295512030827155426342800, 3.91999209839219618951600339672, 5.07745693546192406001256653641, 7.19215859769960394694719084552, 7.70371940499220162259886145654, 8.980390127410147176838644704082, 9.811987655781429602284330059793, 10.64397095621113598250931508781, 12.30386006597575755395880678612

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