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Properties

Label 2-39-39.29-c4-0-10
Degree $2$
Conductor $39$
Sign $0.491 + 0.870i$
Analytic cond. $4.03142$
Root an. cond. $2.00784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.83 + 1.63i)2-s + (6.59 − 6.12i)3-s + (−2.64 + 4.58i)4-s − 28.1i·5-s + (−8.66 + 28.1i)6-s + (−1.35 + 2.35i)7-s − 69.6i·8-s + (5.95 − 80.7i)9-s + (46.0 + 79.7i)10-s + (125. − 72.2i)11-s + (10.6 + 46.4i)12-s + (81.4 − 148. i)13-s − 8.88i·14-s + (−172. − 185. i)15-s + (71.6 + 124. i)16-s + (−68.4 − 39.4i)17-s + ⋯
L(s)  = 1  + (−0.708 + 0.408i)2-s + (0.732 − 0.680i)3-s + (−0.165 + 0.286i)4-s − 1.12i·5-s + (−0.240 + 0.781i)6-s + (−0.0277 + 0.0480i)7-s − 1.08i·8-s + (0.0735 − 0.997i)9-s + (0.460 + 0.797i)10-s + (1.03 − 0.596i)11-s + (0.0738 + 0.322i)12-s + (0.482 − 0.876i)13-s − 0.0453i·14-s + (−0.766 − 0.825i)15-s + (0.279 + 0.484i)16-s + (−0.236 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.491 + 0.870i$
Analytic conductor: \(4.03142\)
Root analytic conductor: \(2.00784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :2),\ 0.491 + 0.870i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.995813 - 0.581588i\)
\(L(\frac12)\) \(\approx\) \(0.995813 - 0.581588i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.59 + 6.12i)T \)
13 \( 1 + (-81.4 + 148. i)T \)
good2 \( 1 + (2.83 - 1.63i)T + (8 - 13.8i)T^{2} \)
5 \( 1 + 28.1iT - 625T^{2} \)
7 \( 1 + (1.35 - 2.35i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-125. + 72.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
17 \( 1 + (68.4 + 39.4i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (237. - 410. i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (553. - 319. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (276. - 159. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 1.75e3T + 9.23e5T^{2} \)
37 \( 1 + (-157. - 273. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-405. + 234. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.00e3 - 1.73e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + 2.37e3iT - 4.87e6T^{2} \)
53 \( 1 - 694. iT - 7.89e6T^{2} \)
59 \( 1 + (-1.65e3 - 955. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.08e3 + 3.61e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.32e3 + 2.28e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.99e3 - 4.61e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 - 5.37e3T + 2.83e7T^{2} \)
79 \( 1 + 1.08e3T + 3.89e7T^{2} \)
83 \( 1 + 2.64e3iT - 4.74e7T^{2} \)
89 \( 1 + (4.86e3 - 2.81e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (2.24e3 - 3.89e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.51543606028287932724440419637, −13.93672299693553016927461310914, −12.92546852872293052462830245500, −12.04229288858064760701064375398, −9.706509718639745068021033300465, −8.581257541879161330920916440571, −8.043981359389640307507115114515, −6.31329291849635734388612290853, −3.80365947221695357024053017314, −1.00324355699227614730415725708, 2.27655889269699503906992829346, 4.28289275965255032287370624597, 6.65639608814058108385552555733, 8.501401938672768936191761574064, 9.559497718478032836847090991149, 10.51569934340052050340104435559, 11.50829007277346388138926369093, 13.78033493225643234399787797033, 14.52677224197776771913498096281, 15.41057936432280936901485447784

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