Properties

Label 2-138-23.13-c1-0-0
Degree $2$
Conductor $138$
Sign $-0.356 - 0.934i$
Analytic cond. $1.10193$
Root an. cond. $1.04973$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−1.61 + 3.53i)5-s + (0.142 + 0.989i)6-s + (−3.99 − 1.17i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (3.72 − 1.09i)10-s + (−2.96 + 3.42i)11-s + (0.654 − 0.755i)12-s + (3.13 − 0.919i)13-s + (1.72 + 3.78i)14-s + (3.26 − 2.10i)15-s + (−0.959 − 0.281i)16-s + (0.260 + 1.81i)17-s + ⋯
L(s)  = 1  + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.721 + 1.58i)5-s + (0.0580 + 0.404i)6-s + (−1.50 − 0.443i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (1.17 − 0.346i)10-s + (−0.894 + 1.03i)11-s + (0.189 − 0.218i)12-s + (0.868 − 0.254i)13-s + (0.462 + 1.01i)14-s + (0.844 − 0.542i)15-s + (−0.239 − 0.0704i)16-s + (0.0631 + 0.438i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.356 - 0.934i$
Analytic conductor: \(1.10193\)
Root analytic conductor: \(1.04973\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1/2),\ -0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165778 + 0.240778i\)
\(L(\frac12)\) \(\approx\) \(0.165778 + 0.240778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (2.28 + 4.21i)T \)
good5 \( 1 + (1.61 - 3.53i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (3.99 + 1.17i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (2.96 - 3.42i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-3.13 + 0.919i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.260 - 1.81i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.194 - 1.35i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.0798 + 0.555i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (3.90 - 2.50i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-1.04 - 2.29i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (2.54 - 5.56i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-8.28 - 5.32i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 2.38T + 47T^{2} \)
53 \( 1 + (10.1 + 2.98i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-6.86 + 2.01i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (2.73 - 1.76i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-4.11 - 4.75i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-4.22 - 4.87i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.75 - 12.2i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (8.09 - 2.37i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (0.156 + 0.342i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-12.7 - 8.22i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-7.84 + 17.1i)T + (-63.5 - 73.3i)T^{2} \)
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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

−13.09506029390124506021882537665, −12.48982307219212807652670968403, −11.22854014276878936101528294283, −10.45500464611209806693496389152, −9.892940593032381709414526611713, −8.017723961279178555268817994232, −7.06193696013240410393145424056, −6.26768665191951009595194230546, −3.90856011211871057868797474926, −2.76247369260552375834314944951, 0.35217438654691582087542550619, 3.71165325706880468394763523266, 5.29038655111059805249455139383, 6.08840828342920896233376312486, 7.68113844721072914269821370069, 8.877177663877024821242207252303, 9.363566809292221826844103420654, 10.77449205439342677119551259362, 11.91537949550447083869699917230, 12.89058700999390160614011658211

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