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Properties

Label 2-138-3.2-c2-0-6
Degree $2$
Conductor $138$
Sign $0.435 - 0.900i$
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 + (1.30 − 2.70i)3-s − 2.00·4-s + 8.14i·5-s + (3.81 + 1.84i)6-s + 7.21·7-s − 2.82i·8-s + (−5.58 − 7.06i)9-s − 11.5·10-s + 12.7i·11-s + (−2.61 + 5.40i)12-s + 12.5·13-s + 10.1i·14-s + (21.9 + 10.6i)15-s + 4.00·16-s + 8.31i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.435 − 0.900i)3-s − 0.500·4-s + 1.62i·5-s + (0.636 + 0.308i)6-s + 1.03·7-s − 0.353i·8-s + (−0.620 − 0.784i)9-s − 1.15·10-s + 1.15i·11-s + (−0.217 + 0.450i)12-s + 0.965·13-s + 0.728i·14-s + (1.46 + 0.709i)15-s + 0.250·16-s + 0.488i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.435 - 0.900i$
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: \((2,\ 138,\ (\ :1),\ 0.435 - 0.900i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40601 + 0.881316i\)
\(L(\frac12)\) \(\approx\) \(1.40601 + 0.881316i\)
\(L(2)\) 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 \( 1 + (-1.30 + 2.70i)T \)
23 \( 1 - 4.79iT \)
good5 \( 1 - 8.14iT - 25T^{2} \)
7 \( 1 - 7.21T + 49T^{2} \)
11 \( 1 - 12.7iT - 121T^{2} \)
13 \( 1 - 12.5T + 169T^{2} \)
17 \( 1 - 8.31iT - 289T^{2} \)
19 \( 1 - 11.8T + 361T^{2} \)
29 \( 1 + 55.7iT - 841T^{2} \)
31 \( 1 + 7.20T + 961T^{2} \)
37 \( 1 + 51.7T + 1.36e3T^{2} \)
41 \( 1 + 74.7iT - 1.68e3T^{2} \)
43 \( 1 - 10.7T + 1.84e3T^{2} \)
47 \( 1 + 66.8iT - 2.20e3T^{2} \)
53 \( 1 - 42.1iT - 2.80e3T^{2} \)
59 \( 1 - 44.3iT - 3.48e3T^{2} \)
61 \( 1 - 77.1T + 3.72e3T^{2} \)
67 \( 1 - 1.87T + 4.48e3T^{2} \)
71 \( 1 + 6.18iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 + 104.T + 6.24e3T^{2} \)
83 \( 1 + 30.8iT - 6.88e3T^{2} \)
89 \( 1 - 1.60iT - 7.92e3T^{2} \)
97 \( 1 + 77.2T + 9.40e3T^{2} \)
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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

−13.59378386341878157415829035557, −12.14871975410619393539430951570, −11.16643596541360582653928657584, −10.00515096606352927402457775312, −8.537212962344187132920760787773, −7.52843621951308285052699299841, −6.91812398515183318202815521160, −5.76605609616234600705052409763, −3.77867512249468028492605846157, −2.05868467223899689720509111862, 1.29164944013677548800870183192, 3.43645231929743092916319113755, 4.74468062897524918128683914403, 5.43023983913461615367081334868, 8.209092685929213496134640227503, 8.636617232569427326546871986034, 9.506685080590810776559246702884, 10.90838558523732386891604455294, 11.50978731961608715299886265380, 12.78232689026551532619821392727

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