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Properties

Label 2-138-69.68-c5-0-15
Degree $2$
Conductor $138$
Sign $0.108 - 0.994i$
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 + (−1.79 − 15.4i)3-s − 16·4-s + 69.6·5-s + (61.9 − 7.18i)6-s + 161. i·7-s − 64i·8-s + (−236. + 55.6i)9-s + 278. i·10-s − 206.·11-s + (28.7 + 247. i)12-s + 54.3·13-s − 646.·14-s + (−124. − 1.07e3i)15-s + 256·16-s + 1.07e3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.115 − 0.993i)3-s − 0.5·4-s + 1.24·5-s + (0.702 − 0.0814i)6-s + 1.24i·7-s − 0.353i·8-s + (−0.973 + 0.228i)9-s + 0.880i·10-s − 0.514·11-s + (0.0575 + 0.496i)12-s + 0.0892·13-s − 0.882·14-s + (−0.143 − 1.23i)15-s + 0.250·16-s + 0.906·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.108 - 0.994i$
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: \((2,\ 138,\ (\ :5/2),\ 0.108 - 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.821285424\)
\(L(\frac12)\) \(\approx\) \(1.821285424\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.79 + 15.4i)T \)
23 \( 1 + (563. - 2.47e3i)T \)
good5 \( 1 - 69.6T + 3.12e3T^{2} \)
7 \( 1 - 161. iT - 1.68e4T^{2} \)
11 \( 1 + 206.T + 1.61e5T^{2} \)
13 \( 1 - 54.3T + 3.71e5T^{2} \)
17 \( 1 - 1.07e3T + 1.41e6T^{2} \)
19 \( 1 - 503. iT - 2.47e6T^{2} \)
29 \( 1 - 7.25e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.83e3T + 2.86e7T^{2} \)
37 \( 1 - 2.90e3iT - 6.93e7T^{2} \)
41 \( 1 - 86.4iT - 1.15e8T^{2} \)
43 \( 1 + 7.00e3iT - 1.47e8T^{2} \)
47 \( 1 - 6.59e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.21e4T + 4.18e8T^{2} \)
59 \( 1 - 2.54e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.41e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.06e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.03e3iT - 1.80e9T^{2} \)
73 \( 1 - 1.08e4T + 2.07e9T^{2} \)
79 \( 1 - 6.77e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.24e4T + 3.93e9T^{2} \)
89 \( 1 - 6.70e4T + 5.58e9T^{2} \)
97 \( 1 + 1.65e5iT - 8.58e9T^{2} \)
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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

−12.68015226349149338971382867128, −11.83430395009098566315320550347, −10.24624017120043725199804691793, −9.124187591729103834797907441297, −8.217048301530464540105119381195, −6.99114754745463256105055989493, −5.72277567165290232326350370248, −5.49743832894772526856754652415, −2.78157244092668216989485763833, −1.46678102169089989737356463538, 0.66975148323843814182198689499, 2.48910245386606791503297163471, 3.91649478248936371107998286911, 5.05485225216334635214806526755, 6.24169112191364866590643251540, 8.034240381287429166271997257991, 9.410559343606935044549919217386, 10.18196957146640458198431541213, 10.58690855813679892158667765116, 11.82507799801443324802003938969

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