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Properties

Label 2-138-69.68-c5-0-9
Degree $2$
Conductor $138$
Sign $0.899 + 0.437i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.899 + 0.437i$
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: \((2,\ 138,\ (\ :5/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5911869172\)
\(L(\frac12)\) \(\approx\) \(0.5911869172\)
\(L(\frac{7}{2})\) 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

−11.99753507506323071670610365711, −11.22058786675896398032217983431, −10.77521069366364131062252652889, −8.834907266015295098404041333111, −7.994287534471492680015917768698, −6.91952539432494503140673101471, −5.39379094390946039393383439789, −4.04749843813076155984479090514, −2.52424996965311114304441986617, −0.75377464519272691334973869219, 0.35519017810405513424898607735, 3.53459014867053200690765738325, 4.40976976931895051558192551715, 5.63764300785131165453466441213, 6.90337851538399887216883809176, 8.171267001020415826461179695130, 8.918370947334093521578477065790, 10.57544484084008316641475420709, 11.04996823946677579645561710300, 12.29824646980894830534378065697

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