↑

Properties

Label 2-138-69.68-c5-0-28
Degree $2$
Conductor $138$
Sign $0.479 - 0.877i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (11.5 + 10.4i)3-s − 16·4-s + 27.4·5-s + (−41.9 + 46.1i)6-s − 207. i·7-s − 64i·8-s + (23.4 + 241. i)9-s + 109. i·10-s + 209.·11-s + (−184. − 167. i)12-s + 802.·13-s + 831.·14-s + (316. + 287. i)15-s + 256·16-s + 869.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.740 + 0.672i)3-s − 0.5·4-s + 0.490·5-s + (−0.475 + 0.523i)6-s − 1.60i·7-s − 0.353i·8-s + (0.0963 + 0.995i)9-s + 0.347i·10-s + 0.522·11-s + (−0.370 − 0.336i)12-s + 1.31·13-s + 1.13·14-s + (0.363 + 0.330i)15-s + 0.250·16-s + 0.729·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.479 - 0.877i$
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: \((2,\ 138,\ (\ :5/2),\ 0.479 - 0.877i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.818187446\)
\(L(\frac12)\) \(\approx\) \(2.818187446\)
\(L(\frac{7}{2})\) 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 \( 1 + (-11.5 - 10.4i)T \)
23 \( 1 + (-2.46e3 + 597. i)T \)
good5 \( 1 - 27.4T + 3.12e3T^{2} \)
7 \( 1 + 207. iT - 1.68e4T^{2} \)
11 \( 1 - 209.T + 1.61e5T^{2} \)
13 \( 1 - 802.T + 3.71e5T^{2} \)
17 \( 1 - 869.T + 1.41e6T^{2} \)
19 \( 1 - 968. iT - 2.47e6T^{2} \)
29 \( 1 - 109. iT - 2.05e7T^{2} \)
31 \( 1 - 1.93e3T + 2.86e7T^{2} \)
37 \( 1 + 6.53e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.17e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.32e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.05e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.36e4T + 4.18e8T^{2} \)
59 \( 1 - 1.80e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.37e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.63e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.69e3iT - 1.80e9T^{2} \)
73 \( 1 - 6.80e4T + 2.07e9T^{2} \)
79 \( 1 + 2.02e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.20e4T + 3.93e9T^{2} \)
89 \( 1 + 5.66e4T + 5.58e9T^{2} \)
97 \( 1 + 1.54e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.93055766394126206092650157560, −11.02387954331647179895500908930, −10.19790053430842976011499201378, −9.283644413845923140487981975561, −8.169121722579968877410944381358, −7.19867612102681384827463498746, −5.87535443116075923263805995349, −4.34572589235671430311957900689, −3.50268823153881461465146709777, −1.21530802217597149531067426122, 1.24451237531272202870764362230, 2.38430426383489563539617861014, 3.50421366405777462005737979676, 5.47122710242166043030566450797, 6.54656900178012973360027137576, 8.278310249308312501059311956138, 8.949305885735982641100804159326, 9.769272975067807231342092560721, 11.39023942401290830483473672783, 12.10587539777568224025726665584

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