Properties

Label 2-138-69.68-c5-0-22
Degree $2$
Conductor $138$
Sign $0.0274 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (−8.44 + 13.0i)3-s − 16·4-s + 77.2·5-s + (−52.3 − 33.7i)6-s + 18.1i·7-s − 64i·8-s + (−100. − 221. i)9-s + 309. i·10-s + 470.·11-s + (135. − 209. i)12-s + 910.·13-s − 72.5·14-s + (−652. + 1.01e3i)15-s + 256·16-s + 1.57e3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.542 + 0.840i)3-s − 0.5·4-s + 1.38·5-s + (−0.594 − 0.383i)6-s + 0.139i·7-s − 0.353i·8-s + (−0.412 − 0.911i)9-s + 0.977i·10-s + 1.17·11-s + (0.271 − 0.420i)12-s + 1.49·13-s − 0.0988·14-s + (−0.749 + 1.16i)15-s + 0.250·16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0274 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0274 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.0274 - 0.999i$
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: \((2,\ 138,\ (\ :5/2),\ 0.0274 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.213390429\)
\(L(\frac12)\) \(\approx\) \(2.213390429\)
\(L(\frac{7}{2})\) 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 \( 1 + (8.44 - 13.0i)T \)
23 \( 1 + (1.31e3 + 2.16e3i)T \)
good5 \( 1 - 77.2T + 3.12e3T^{2} \)
7 \( 1 - 18.1iT - 1.68e4T^{2} \)
11 \( 1 - 470.T + 1.61e5T^{2} \)
13 \( 1 - 910.T + 3.71e5T^{2} \)
17 \( 1 - 1.57e3T + 1.41e6T^{2} \)
19 \( 1 + 1.83e3iT - 2.47e6T^{2} \)
29 \( 1 - 5.11e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.80e3T + 2.86e7T^{2} \)
37 \( 1 - 1.00e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.23e4iT - 1.15e8T^{2} \)
43 \( 1 + 4.49e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.54e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.90e4T + 4.18e8T^{2} \)
59 \( 1 + 6.45e3iT - 7.14e8T^{2} \)
61 \( 1 + 1.48e3iT - 8.44e8T^{2} \)
67 \( 1 - 6.08e3iT - 1.35e9T^{2} \)
71 \( 1 + 4.60e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.70e4T + 2.07e9T^{2} \)
79 \( 1 - 8.81e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.68e4T + 3.93e9T^{2} \)
89 \( 1 - 7.46e4T + 5.58e9T^{2} \)
97 \( 1 + 2.32e4iT - 8.58e9T^{2} \)
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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

−12.62476389506476198050383505232, −11.33744356191081386275102090947, −10.24193174851525889952884566335, −9.368024840749155890570268533599, −8.629764862120247080533443571253, −6.59985115694182137717929894290, −6.00027973564397330776992496272, −4.96209516242695747628806228914, −3.49344201858221029699666223516, −1.18256891922554845868119097242, 1.14060121638196952391272204951, 1.85067049999592233013097510365, 3.70736176686487921170231657409, 5.70868016222913629151728922031, 6.12577625561589363321956701887, 7.76045929888539657606339640748, 9.107018132423762538262453593780, 10.08046585025234518449647236962, 11.08470609192157886989751886989, 12.06548703475070988203224985052

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