Properties

Label 2-39-13.9-c3-0-4
Degree $2$
Conductor $39$
Sign $-0.140 + 0.990i$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 + 4.61i)2-s + (1.5 − 2.59i)3-s + (−10.2 − 17.6i)4-s − 16.4·5-s + (7.99 + 13.8i)6-s + (−4.83 − 8.38i)7-s + 66.1·8-s + (−4.5 − 7.79i)9-s + (43.7 − 75.7i)10-s + (−13.7 + 23.8i)11-s − 61.2·12-s + (−37.3 + 28.3i)13-s + 51.5·14-s + (−24.6 + 42.6i)15-s + (−94.6 + 164. i)16-s + (−53.9 − 93.4i)17-s + ⋯
L(s)  = 1  + (−0.942 + 1.63i)2-s + (0.288 − 0.499i)3-s + (−1.27 − 2.20i)4-s − 1.46·5-s + (0.544 + 0.942i)6-s + (−0.261 − 0.452i)7-s + 2.92·8-s + (−0.166 − 0.288i)9-s + (1.38 − 2.39i)10-s + (−0.378 + 0.654i)11-s − 1.47·12-s + (−0.795 + 0.605i)13-s + 0.984·14-s + (−0.423 + 0.734i)15-s + (−1.47 + 2.56i)16-s + (−0.769 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.140 + 0.990i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.140 + 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0510216 - 0.0588004i\)
\(L(\frac12)\) \(\approx\) \(0.0510216 - 0.0588004i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
13 \( 1 + (37.3 - 28.3i)T \)
good2 \( 1 + (2.66 - 4.61i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 16.4T + 125T^{2} \)
7 \( 1 + (4.83 + 8.38i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (13.7 - 23.8i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (53.9 + 93.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-1.12 - 1.94i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (20.9 - 36.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (30.8 - 53.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 191.T + 2.97e4T^{2} \)
37 \( 1 + (49.2 - 85.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-15.3 + 26.6i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (119. + 206. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 511.T + 1.03e5T^{2} \)
53 \( 1 - 492.T + 1.48e5T^{2} \)
59 \( 1 + (242. + 419. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-222. - 384. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (95.0 - 164. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (242. + 419. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 957.T + 3.89e5T^{2} \)
79 \( 1 + 375.T + 4.93e5T^{2} \)
83 \( 1 + 715.T + 5.71e5T^{2} \)
89 \( 1 + (-519. + 899. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (32.7 + 56.7i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.61200861426301481538488809880, −14.74616870473066127623969539287, −13.52972756094607674840566234158, −11.74932812467048441481655632215, −9.931332746800595128937204147442, −8.612909119152836420883401168369, −7.43192465813872438502670614890, −6.94683571580237836399208664880, −4.68977682605091445267589241218, −0.083897540726806322691713104340, 2.90802255310925398658315571229, 4.21025688558061772594676090347, 7.943361062328795225967608990626, 8.660046474197514240549301214800, 10.12055904643293849596259285172, 11.11043745353314266344844145706, 12.08110943548292090044952283716, 13.09012816121291161331643203140, 15.09499215344009823328418947049, 16.21738746139724443110943171745

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