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Properties

Label 2-138-69.68-c5-0-18
Degree $2$
Conductor $138$
Sign $-0.249 - 0.968i$
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 + (15.5 − 0.805i)3-s − 16·4-s + 10.2·5-s + (3.22 + 62.2i)6-s + 139. i·7-s − 64i·8-s + (241. − 25.0i)9-s + 41.1i·10-s + 570.·11-s + (−249. + 12.8i)12-s − 754.·13-s − 559.·14-s + (160. − 8.29i)15-s + 256·16-s + 143.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.998 − 0.0516i)3-s − 0.5·4-s + 0.184·5-s + (0.0365 + 0.706i)6-s + 1.07i·7-s − 0.353i·8-s + (0.994 − 0.103i)9-s + 0.130i·10-s + 1.42·11-s + (−0.499 + 0.0258i)12-s − 1.23·13-s − 0.762·14-s + (0.183 − 0.00951i)15-s + 0.250·16-s + 0.120·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.249 - 0.968i$
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: \((2,\ 138,\ (\ :5/2),\ -0.249 - 0.968i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.679870405\)
\(L(\frac12)\) \(\approx\) \(2.679870405\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.5 + 0.805i)T \)
23 \( 1 + (-2.48e3 + 505. i)T \)
good5 \( 1 - 10.2T + 3.12e3T^{2} \)
7 \( 1 - 139. iT - 1.68e4T^{2} \)
11 \( 1 - 570.T + 1.61e5T^{2} \)
13 \( 1 + 754.T + 3.71e5T^{2} \)
17 \( 1 - 143.T + 1.41e6T^{2} \)
19 \( 1 - 2.08e3iT - 2.47e6T^{2} \)
29 \( 1 - 5.65e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.62e3T + 2.86e7T^{2} \)
37 \( 1 - 7.24e3iT - 6.93e7T^{2} \)
41 \( 1 - 965. iT - 1.15e8T^{2} \)
43 \( 1 - 1.20e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.93e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.88e4T + 4.18e8T^{2} \)
59 \( 1 - 2.07e4iT - 7.14e8T^{2} \)
61 \( 1 + 8.37e3iT - 8.44e8T^{2} \)
67 \( 1 - 2.83e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.91e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.89e3T + 2.07e9T^{2} \)
79 \( 1 + 9.91e4iT - 3.07e9T^{2} \)
83 \( 1 - 8.60e4T + 3.93e9T^{2} \)
89 \( 1 - 1.74e4T + 5.58e9T^{2} \)
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

−12.61946971325404446245072264298, −11.91941088827747129373513481568, −9.978110735636600649580395228295, −9.236080177462019598654526191219, −8.413220234969096192262143694585, −7.26923057084186456114887377671, −6.15121309926564725754012779141, −4.72485904345855171424065504138, −3.25284952358138721612173716452, −1.69092439093114845373355548506, 0.887332698181276533211461727728, 2.34027185494983731490168640337, 3.72387166586013676889258417203, 4.66043723680437824122419733787, 6.80804604377739443963519158921, 7.76982318816483241229254925862, 9.234169250757158305000238480724, 9.680134639195813905461690552790, 10.86730498964646688528057657956, 11.99668565313745091990584469416

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