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Properties

Label 2-138-3.2-c2-0-0
Degree $2$
Conductor $138$
Sign $-0.250 - 0.968i$
Analytic cond. $3.76022$
Root an. cond. $1.93913$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (−0.752 − 2.90i)3-s − 2.00·4-s + 9.12i·5-s + (−4.10 + 1.06i)6-s − 11.5·7-s + 2.82i·8-s + (−7.86 + 4.36i)9-s + 12.9·10-s − 3.58i·11-s + (1.50 + 5.80i)12-s − 14.9·13-s + 16.3i·14-s + (26.5 − 6.86i)15-s + 4.00·16-s − 0.210i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.250 − 0.968i)3-s − 0.500·4-s + 1.82i·5-s + (−0.684 + 0.177i)6-s − 1.64·7-s + 0.353i·8-s + (−0.874 + 0.485i)9-s + 1.29·10-s − 0.325i·11-s + (0.125 + 0.484i)12-s − 1.15·13-s + 1.16i·14-s + (1.76 − 0.457i)15-s + 0.250·16-s − 0.0123i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.250 - 0.968i$
Analytic conductor: \(3.76022\)
Root analytic conductor: \(1.93913\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1),\ -0.250 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.130896 + 0.169128i\)
\(L(\frac12)\) \(\approx\) \(0.130896 + 0.169128i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (0.752 + 2.90i)T \)
23 \( 1 + 4.79iT \)
good5 \( 1 - 9.12iT - 25T^{2} \)
7 \( 1 + 11.5T + 49T^{2} \)
11 \( 1 + 3.58iT - 121T^{2} \)
13 \( 1 + 14.9T + 169T^{2} \)
17 \( 1 + 0.210iT - 289T^{2} \)
19 \( 1 - 22.3T + 361T^{2} \)
29 \( 1 + 23.9iT - 841T^{2} \)
31 \( 1 + 40.1T + 961T^{2} \)
37 \( 1 + 14.3T + 1.36e3T^{2} \)
41 \( 1 - 41.9iT - 1.68e3T^{2} \)
43 \( 1 + 18.5T + 1.84e3T^{2} \)
47 \( 1 - 21.5iT - 2.20e3T^{2} \)
53 \( 1 - 72.8iT - 2.80e3T^{2} \)
59 \( 1 + 55.5iT - 3.48e3T^{2} \)
61 \( 1 - 74.4T + 3.72e3T^{2} \)
67 \( 1 + 94.2T + 4.48e3T^{2} \)
71 \( 1 - 62.3iT - 5.04e3T^{2} \)
73 \( 1 + 60.5T + 5.32e3T^{2} \)
79 \( 1 - 51.1T + 6.24e3T^{2} \)
83 \( 1 - 5.69iT - 6.88e3T^{2} \)
89 \( 1 + 41.8iT - 7.92e3T^{2} \)
97 \( 1 + 61.6T + 9.40e3T^{2} \)
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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

−13.16333058191758676847908877646, −12.15892692583414589389453474568, −11.30893446644357522011246801132, −10.27887950769496724586404507872, −9.483633634542321574019957224901, −7.58298609814659563971655886597, −6.81143630959009117624296243322, −5.82926025822172625086624448219, −3.31585927713845861644829930811, −2.56765892801447976636322267669, 0.13789146882765524519970282587, 3.63715724831175664234457583239, 4.95507567393907087408651545351, 5.65371501361686312635941005260, 7.20725585501594956898838061306, 8.783017509843311831028700352254, 9.439872870256033849989780486897, 10.04262372571789566825113288119, 12.00646555120400909576692252051, 12.64349624631638879585929015354

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