Properties

Label 2-138-23.22-c2-0-0
Degree $2$
Conductor $138$
Sign $-0.194 - 0.980i$
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  − 1.41·2-s − 1.73·3-s + 2.00·4-s + 1.69i·5-s + 2.44·6-s − 4.18i·7-s − 2.82·8-s + 2.99·9-s − 2.39i·10-s + 20.2i·11-s − 3.46·12-s − 11.6·13-s + 5.91i·14-s − 2.93i·15-s + 4.00·16-s + 12.5i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.338i·5-s + 0.408·6-s − 0.597i·7-s − 0.353·8-s + 0.333·9-s − 0.239i·10-s + 1.84i·11-s − 0.288·12-s − 0.895·13-s + 0.422i·14-s − 0.195i·15-s + 0.250·16-s + 0.737i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.194 - 0.980i$
Analytic conductor: \(3.76022\)
Root analytic conductor: \(1.93913\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1),\ -0.194 - 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.385792 + 0.469710i\)
\(L(\frac12)\) \(\approx\) \(0.385792 + 0.469710i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + 1.73T \)
23 \( 1 + (-4.46 - 22.5i)T \)
good5 \( 1 - 1.69iT - 25T^{2} \)
7 \( 1 + 4.18iT - 49T^{2} \)
11 \( 1 - 20.2iT - 121T^{2} \)
13 \( 1 + 11.6T + 169T^{2} \)
17 \( 1 - 12.5iT - 289T^{2} \)
19 \( 1 - 27.1iT - 361T^{2} \)
29 \( 1 + 50.9T + 841T^{2} \)
31 \( 1 + 11.5T + 961T^{2} \)
37 \( 1 + 52.7iT - 1.36e3T^{2} \)
41 \( 1 - 42.4T + 1.68e3T^{2} \)
43 \( 1 - 2.10iT - 1.84e3T^{2} \)
47 \( 1 - 20.0T + 2.20e3T^{2} \)
53 \( 1 + 49.6iT - 2.80e3T^{2} \)
59 \( 1 + 39.9T + 3.48e3T^{2} \)
61 \( 1 - 70.9iT - 3.72e3T^{2} \)
67 \( 1 - 4.38iT - 4.48e3T^{2} \)
71 \( 1 + 46.5T + 5.04e3T^{2} \)
73 \( 1 + 3.71T + 5.32e3T^{2} \)
79 \( 1 - 26.4iT - 6.24e3T^{2} \)
83 \( 1 - 106. iT - 6.88e3T^{2} \)
89 \( 1 + 137. iT - 7.92e3T^{2} \)
97 \( 1 - 119. iT - 9.40e3T^{2} \)
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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

−12.84631185943815576055848660343, −12.20321473577169370388700228483, −10.95288166633536683539539423574, −10.15439715695436450728543224542, −9.386108466282130351181569511762, −7.55714754757754878704732813533, −7.15197108998718890466051704686, −5.62528613372316216337428902055, −4.06120813895957730961440368010, −1.85329345049316276159532135496, 0.53753460350578669345141299914, 2.81001626623434818907116674603, 4.97493044292918535635785041907, 6.08571610984731853430625547356, 7.32296328140980739874693151859, 8.686056617057109691603162870747, 9.338390888101283518508353463457, 10.80253683566701502815724453335, 11.41764449897232193353961741878, 12.44639088320610604529365047933

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