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Properties

Label 2-138-69.68-c3-0-17
Degree $2$
Conductor $138$
Sign $0.502 + 0.864i$
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 + (4.90 + 1.72i)3-s − 4·4-s + 10.2·5-s + (3.44 − 9.80i)6-s − 24.1i·7-s + 8i·8-s + (21.0 + 16.8i)9-s − 20.5i·10-s − 49.0·11-s + (−19.6 − 6.88i)12-s + 80.1·13-s − 48.3·14-s + (50.4 + 17.7i)15-s + 16·16-s + 96.2·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.943 + 0.331i)3-s − 0.5·4-s + 0.919·5-s + (0.234 − 0.667i)6-s − 1.30i·7-s + 0.353i·8-s + (0.780 + 0.625i)9-s − 0.650i·10-s − 1.34·11-s + (−0.471 − 0.165i)12-s + 1.71·13-s − 0.923·14-s + (0.867 + 0.304i)15-s + 0.250·16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.502 + 0.864i$
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: \((2,\ 138,\ (\ :3/2),\ 0.502 + 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.12850 - 1.22556i\)
\(L(\frac12)\) \(\approx\) \(2.12850 - 1.22556i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.90 - 1.72i)T \)
23 \( 1 + (71.6 - 83.8i)T \)
good5 \( 1 - 10.2T + 125T^{2} \)
7 \( 1 + 24.1iT - 343T^{2} \)
11 \( 1 + 49.0T + 1.33e3T^{2} \)
13 \( 1 - 80.1T + 2.19e3T^{2} \)
17 \( 1 - 96.2T + 4.91e3T^{2} \)
19 \( 1 + 82.7iT - 6.85e3T^{2} \)
29 \( 1 + 174. iT - 2.43e4T^{2} \)
31 \( 1 + 119.T + 2.97e4T^{2} \)
37 \( 1 - 361. iT - 5.06e4T^{2} \)
41 \( 1 - 57.8iT - 6.89e4T^{2} \)
43 \( 1 + 314. iT - 7.95e4T^{2} \)
47 \( 1 - 364. iT - 1.03e5T^{2} \)
53 \( 1 + 284.T + 1.48e5T^{2} \)
59 \( 1 - 130. iT - 2.05e5T^{2} \)
61 \( 1 - 613. iT - 2.26e5T^{2} \)
67 \( 1 - 922. iT - 3.00e5T^{2} \)
71 \( 1 + 209. iT - 3.57e5T^{2} \)
73 \( 1 + 693.T + 3.89e5T^{2} \)
79 \( 1 - 881. iT - 4.93e5T^{2} \)
83 \( 1 + 437.T + 5.71e5T^{2} \)
89 \( 1 + 750.T + 7.04e5T^{2} \)
97 \( 1 + 438. iT - 9.12e5T^{2} \)
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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

−13.02383875150740753172596340827, −11.20390633184545103770735488769, −10.24786044639764748716577256030, −9.808399467834872442312433291719, −8.450212072984493656596726362654, −7.49357305519732311913791807107, −5.65600384481116420917915906497, −4.14164813824504819302832109351, −2.98276250301682562848627875293, −1.37738351464054877547968095916, 1.87182179167766022411110662646, 3.37986948958869012715179271332, 5.49586961202101866299103988561, 6.14897765418425336667554422513, 7.80309043475952128933799426367, 8.528444673248043710687400990010, 9.455261102267034023237597258353, 10.50466621530808362209059510716, 12.39588075899954368432532779632, 13.02429670690241761568105873894

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