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Properties

Label 2-138-69.68-c3-0-10
Degree $2$
Conductor $138$
Sign $0.677 + 0.735i$
Analytic cond. $8.14226$
Root an. cond. $2.85346$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + (−3.26 + 4.04i)3-s − 4·4-s − 6.82·5-s + (8.08 + 6.53i)6-s − 3.09i·7-s + 8i·8-s + (−5.67 − 26.3i)9-s + 13.6i·10-s + 50.4·11-s + (13.0 − 16.1i)12-s + 6.18·13-s − 6.19·14-s + (22.2 − 27.5i)15-s + 16·16-s + 118.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.628 + 0.777i)3-s − 0.5·4-s − 0.610·5-s + (0.550 + 0.444i)6-s − 0.167i·7-s + 0.353i·8-s + (−0.210 − 0.977i)9-s + 0.431i·10-s + 1.38·11-s + (0.314 − 0.388i)12-s + 0.131·13-s − 0.118·14-s + (0.383 − 0.474i)15-s + 0.250·16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(8.14226\)
Root analytic conductor: \(2.85346\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06772 - 0.467834i\)
\(L(\frac12)\) \(\approx\) \(1.06772 - 0.467834i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 + (3.26 - 4.04i)T \)
23 \( 1 + (-109. - 16.0i)T \)
good5 \( 1 + 6.82T + 125T^{2} \)
7 \( 1 + 3.09iT - 343T^{2} \)
11 \( 1 - 50.4T + 1.33e3T^{2} \)
13 \( 1 - 6.18T + 2.19e3T^{2} \)
17 \( 1 - 118.T + 4.91e3T^{2} \)
19 \( 1 + 92.8iT - 6.85e3T^{2} \)
29 \( 1 + 108. iT - 2.43e4T^{2} \)
31 \( 1 + 49.4T + 2.97e4T^{2} \)
37 \( 1 - 55.2iT - 5.06e4T^{2} \)
41 \( 1 - 136. iT - 6.89e4T^{2} \)
43 \( 1 + 211. iT - 7.95e4T^{2} \)
47 \( 1 + 451. iT - 1.03e5T^{2} \)
53 \( 1 - 265.T + 1.48e5T^{2} \)
59 \( 1 - 653. iT - 2.05e5T^{2} \)
61 \( 1 - 687. iT - 2.26e5T^{2} \)
67 \( 1 + 10.5iT - 3.00e5T^{2} \)
71 \( 1 + 314. iT - 3.57e5T^{2} \)
73 \( 1 - 795.T + 3.89e5T^{2} \)
79 \( 1 + 1.22e3iT - 4.93e5T^{2} \)
83 \( 1 + 403.T + 5.71e5T^{2} \)
89 \( 1 + 29.6T + 7.04e5T^{2} \)
97 \( 1 + 1.16e3iT - 9.12e5T^{2} \)
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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

−11.96873600729307561494878574007, −11.75377960898457052214017453382, −10.64093652328799205360216126851, −9.653802715499711210122166791022, −8.755893187795375930538259996521, −7.14344094954187067221864151602, −5.64999373677644632799715423200, −4.31442515718108495565151913366, −3.39596029013833463290608256601, −0.844573275716918944875895978131, 1.16392806854821933556105398400, 3.73446598693078133542716158834, 5.34258221092590706476102124709, 6.36500111113648268799609625047, 7.41196191405831617815097738903, 8.264411164031218237909126893734, 9.592797999081101567156101546547, 11.03173526867190874996693757636, 12.08903507693142382818716494050, 12.61732917710364023392044439057

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