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Properties

Label 2-138-69.68-c5-0-31
Degree $2$
Conductor $138$
Sign $0.0461 + 0.998i$
Analytic cond. $22.1329$
Root an. cond. $4.70456$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (6.80 − 14.0i)3-s − 16·4-s − 22.5·5-s + (56.1 + 27.2i)6-s + 106. i·7-s − 64i·8-s + (−150. − 190. i)9-s − 90.1i·10-s + 443.·11-s + (−108. + 224. i)12-s − 103.·13-s − 426.·14-s + (−153. + 316. i)15-s + 256·16-s + 387.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.436 − 0.899i)3-s − 0.5·4-s − 0.403·5-s + (0.636 + 0.308i)6-s + 0.821i·7-s − 0.353i·8-s + (−0.619 − 0.785i)9-s − 0.285i·10-s + 1.10·11-s + (−0.218 + 0.449i)12-s − 0.170·13-s − 0.580·14-s + (−0.175 + 0.362i)15-s + 0.250·16-s + 0.325·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0461 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0461 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.0461 + 0.998i$
Analytic conductor: \(22.1329\)
Root analytic conductor: \(4.70456\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :5/2),\ 0.0461 + 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.308128393\)
\(L(\frac12)\) \(\approx\) \(1.308128393\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 + (-6.80 + 14.0i)T \)
23 \( 1 + (1.21e3 + 2.22e3i)T \)
good5 \( 1 + 22.5T + 3.12e3T^{2} \)
7 \( 1 - 106. iT - 1.68e4T^{2} \)
11 \( 1 - 443.T + 1.61e5T^{2} \)
13 \( 1 + 103.T + 3.71e5T^{2} \)
17 \( 1 - 387.T + 1.41e6T^{2} \)
19 \( 1 + 3.03e3iT - 2.47e6T^{2} \)
29 \( 1 + 4.53e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.72e3T + 2.86e7T^{2} \)
37 \( 1 + 1.33e4iT - 6.93e7T^{2} \)
41 \( 1 + 9.03e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.05e4iT - 1.47e8T^{2} \)
47 \( 1 + 4.42e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.62e4T + 4.18e8T^{2} \)
59 \( 1 + 3.47e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.01e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.48e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.38e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.38e4T + 2.07e9T^{2} \)
79 \( 1 + 2.00e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.51e4T + 3.93e9T^{2} \)
89 \( 1 + 1.25e5T + 5.58e9T^{2} \)
97 \( 1 + 8.95e4iT - 8.58e9T^{2} \)
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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

−12.18801595032320877676160378963, −11.33390470218296544580162106472, −9.361904578493550727419984279285, −8.777319583640553190323088320694, −7.62387669190272311814142283823, −6.71972379172987309876356937650, −5.64722706412286072968136511788, −3.94919752804033105416288886242, −2.29666461711162121367039734626, −0.43739263939224550070633872656, 1.57939801611472432692841858556, 3.58657803881354270405544508769, 3.99337614313937471522728782329, 5.56101954178740123043757589171, 7.44043357474376135840292877080, 8.536318986673565294035111610894, 9.680954298247839289558268196705, 10.34272576127764987690918548354, 11.41560122679576570663427686127, 12.26891997681889466997788489034

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