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Properties

Label 2-138-3.2-c2-0-13
Degree $2$
Conductor $138$
Sign $-0.542 + 0.840i$
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 + (1.62 − 2.52i)3-s − 2.00·4-s − 1.81i·5-s + (−3.56 − 2.30i)6-s + 9.47·7-s + 2.82i·8-s + (−3.70 − 8.20i)9-s − 2.56·10-s − 11.4i·11-s + (−3.25 + 5.04i)12-s − 20.7·13-s − 13.3i·14-s + (−4.56 − 2.94i)15-s + 4.00·16-s + 27.7i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.542 − 0.840i)3-s − 0.500·4-s − 0.362i·5-s + (−0.593 − 0.383i)6-s + 1.35·7-s + 0.353i·8-s + (−0.411 − 0.911i)9-s − 0.256·10-s − 1.04i·11-s + (−0.271 + 0.420i)12-s − 1.59·13-s − 0.957i·14-s + (−0.304 − 0.196i)15-s + 0.250·16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.542 + 0.840i$
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: \((2,\ 138,\ (\ :1),\ -0.542 + 0.840i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.795897 - 1.46150i\)
\(L(\frac12)\) \(\approx\) \(0.795897 - 1.46150i\)
\(L(2)\) 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 \( 1 + (-1.62 + 2.52i)T \)
23 \( 1 - 4.79iT \)
good5 \( 1 + 1.81iT - 25T^{2} \)
7 \( 1 - 9.47T + 49T^{2} \)
11 \( 1 + 11.4iT - 121T^{2} \)
13 \( 1 + 20.7T + 169T^{2} \)
17 \( 1 - 27.7iT - 289T^{2} \)
19 \( 1 - 10.5T + 361T^{2} \)
29 \( 1 + 20.5iT - 841T^{2} \)
31 \( 1 - 13.1T + 961T^{2} \)
37 \( 1 - 63.2T + 1.36e3T^{2} \)
41 \( 1 - 1.80iT - 1.68e3T^{2} \)
43 \( 1 - 47.2T + 1.84e3T^{2} \)
47 \( 1 - 48.8iT - 2.20e3T^{2} \)
53 \( 1 + 19.1iT - 2.80e3T^{2} \)
59 \( 1 - 117. iT - 3.48e3T^{2} \)
61 \( 1 + 63.4T + 3.72e3T^{2} \)
67 \( 1 + 87.8T + 4.48e3T^{2} \)
71 \( 1 + 6.81iT - 5.04e3T^{2} \)
73 \( 1 - 74.4T + 5.32e3T^{2} \)
79 \( 1 + 28.6T + 6.24e3T^{2} \)
83 \( 1 - 20.7iT - 6.88e3T^{2} \)
89 \( 1 - 23.5iT - 7.92e3T^{2} \)
97 \( 1 - 24.5T + 9.40e3T^{2} \)
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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

−12.51950949272744553366009944153, −11.75505766662407622293814273279, −10.77210311178661218956636787325, −9.353076382957052267537697451960, −8.306072623204058782086076617165, −7.65318251506124445595749165976, −5.84134715895347701522019339356, −4.40224759903768866522885311686, −2.65374539442180668592030008237, −1.21520573525187696533539902187, 2.59330411397690921537343804337, 4.64759145220839506656723275847, 5.05283791834964826896983766396, 7.20413135716222088375825479453, 7.81847992927494853224771204334, 9.183941514406778627141022842474, 9.932863680606458681508450460059, 11.13402863396327695812987640943, 12.25970472671766244322407634153, 13.81802111144473704705880719157

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