Properties

Label 2-138-23.22-c6-0-19
Degree $2$
Conductor $138$
Sign $-0.996 + 0.0847i$
Analytic cond. $31.7474$
Root an. cond. $5.63448$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65·2-s + 15.5·3-s + 32.0·4-s − 155. i·5-s − 88.1·6-s + 550. i·7-s − 181.·8-s + 243·9-s + 882. i·10-s − 1.34e3i·11-s + 498.·12-s − 2.41e3·13-s − 3.11e3i·14-s − 2.43e3i·15-s + 1.02e3·16-s − 3.85e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.500·4-s − 1.24i·5-s − 0.408·6-s + 1.60i·7-s − 0.353·8-s + 0.333·9-s + 0.882i·10-s − 1.01i·11-s + 0.288·12-s − 1.09·13-s − 1.13i·14-s − 0.720i·15-s + 0.250·16-s − 0.784i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0847i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 + 0.0847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.996 + 0.0847i$
Analytic conductor: \(31.7474\)
Root analytic conductor: \(5.63448\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3),\ -0.996 + 0.0847i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.3183877056\)
\(L(\frac12)\) \(\approx\) \(0.3183877056\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65T \)
3 \( 1 - 15.5T \)
23 \( 1 + (1.21e4 - 1.03e3i)T \)
good5 \( 1 + 155. iT - 1.56e4T^{2} \)
7 \( 1 - 550. iT - 1.17e5T^{2} \)
11 \( 1 + 1.34e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.41e3T + 4.82e6T^{2} \)
17 \( 1 + 3.85e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.56e3iT - 4.70e7T^{2} \)
29 \( 1 - 2.39e4T + 5.94e8T^{2} \)
31 \( 1 + 1.96e4T + 8.87e8T^{2} \)
37 \( 1 + 1.21e3iT - 2.56e9T^{2} \)
41 \( 1 + 1.32e5T + 4.75e9T^{2} \)
43 \( 1 + 9.59e4iT - 6.32e9T^{2} \)
47 \( 1 + 3.38e4T + 1.07e10T^{2} \)
53 \( 1 - 2.23e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.72e5T + 4.21e10T^{2} \)
61 \( 1 - 1.10e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.40e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.75e5T + 1.28e11T^{2} \)
73 \( 1 + 7.46e5T + 1.51e11T^{2} \)
79 \( 1 + 9.57e5iT - 2.43e11T^{2} \)
83 \( 1 + 6.17e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.09e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.13e6iT - 8.32e11T^{2} \)
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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

−11.98409631825827330257619848688, −10.17693196639083943726491588402, −9.037222020029453010202101269863, −8.734130043606537147824005587430, −7.69521796402511137656258352223, −5.97392966090832876322187159293, −4.92696882477835880483316948476, −2.96262697939700111696386955014, −1.73517127757890787758724072949, −0.10615037371985572059049196879, 1.76525330355891693584320087259, 3.09817654839239165380463634165, 4.41201072658276121450954444686, 6.71631732825406013397591700518, 7.20895583673422213204920421609, 8.131845628011190771080842450587, 9.895781021678274314050201091959, 10.14435871103054921043934954850, 11.15381251845063526828177732773, 12.50909404854895198021156994013

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