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Properties

Label 2-138-69.68-c5-0-23
Degree $2$
Conductor $138$
Sign $-0.732 + 0.681i$
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 + (−15.1 + 3.51i)3-s − 16·4-s − 0.0328·5-s + (14.0 + 60.7i)6-s + 156. i·7-s + 64i·8-s + (218. − 106. i)9-s + 0.131i·10-s − 514.·11-s + (242. − 56.2i)12-s + 415.·13-s + 626.·14-s + (0.498 − 0.115i)15-s + 256·16-s + 1.76e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.974 + 0.225i)3-s − 0.5·4-s − 0.000586·5-s + (0.159 + 0.688i)6-s + 1.20i·7-s + 0.353i·8-s + (0.898 − 0.439i)9-s + 0.000414i·10-s − 1.28·11-s + (0.487 − 0.112i)12-s + 0.682·13-s + 0.853·14-s + (0.000571 − 0.000132i)15-s + 0.250·16-s + 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.681i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.732 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.732 + 0.681i$
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: \((2,\ 138,\ (\ :5/2),\ -0.732 + 0.681i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5400614634\)
\(L(\frac12)\) \(\approx\) \(0.5400614634\)
\(L(\frac{7}{2})\) 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 \( 1 + (15.1 - 3.51i)T \)
23 \( 1 + (1.26e3 + 2.19e3i)T \)
good5 \( 1 + 0.0328T + 3.12e3T^{2} \)
7 \( 1 - 156. iT - 1.68e4T^{2} \)
11 \( 1 + 514.T + 1.61e5T^{2} \)
13 \( 1 - 415.T + 3.71e5T^{2} \)
17 \( 1 - 1.76e3T + 1.41e6T^{2} \)
19 \( 1 - 327. iT - 2.47e6T^{2} \)
29 \( 1 + 1.44e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.82e3T + 2.86e7T^{2} \)
37 \( 1 + 1.69e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.22e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.13e4iT - 1.47e8T^{2} \)
47 \( 1 - 972. iT - 2.29e8T^{2} \)
53 \( 1 - 1.62e4T + 4.18e8T^{2} \)
59 \( 1 + 7.88e3iT - 7.14e8T^{2} \)
61 \( 1 - 3.39e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.62e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.42e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.67e4T + 2.07e9T^{2} \)
79 \( 1 - 1.96e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.52e4T + 3.93e9T^{2} \)
89 \( 1 - 1.16e5T + 5.58e9T^{2} \)
97 \( 1 - 1.26e5iT - 8.58e9T^{2} \)
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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

−12.04333802746289609791792130929, −10.82803005384902265872352431783, −10.13906390779830687832411856856, −8.969620349615472490034668806286, −7.69677534458279893217368642555, −5.83887499631976261347628416200, −5.32095791910404340633279032059, −3.68292577353556814911645443211, −2.05550819164523267487437868102, −0.24545010424134447274998558584, 1.15617893753720668371241409080, 3.76338080024767889029603681336, 5.14151219651634814660507774058, 6.03947843697112596894096878750, 7.39210084252122657512164638590, 7.87855226013900986146656735260, 9.788561086181916931039671727974, 10.54365638814625925759752365703, 11.56777494529696671739617064095, 12.95503061343555430769634722771

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