Properties

Label 2-39-13.12-c3-0-5
Degree $2$
Conductor $39$
Sign $0.159 + 0.987i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.32i·2-s − 3·3-s + 6.23·4-s − 15.4i·5-s + 3.98i·6-s − 7.96i·7-s − 18.9i·8-s + 9·9-s − 20.4·10-s + 12.7i·11-s − 18.7·12-s + (7.47 + 46.2i)13-s − 10.5·14-s + 46.2i·15-s + 24.8·16-s + 54·17-s + ⋯
L(s)  = 1  − 0.469i·2-s − 0.577·3-s + 0.779·4-s − 1.37i·5-s + 0.270i·6-s − 0.430i·7-s − 0.835i·8-s + 0.333·9-s − 0.647·10-s + 0.350i·11-s − 0.450·12-s + (0.159 + 0.987i)13-s − 0.201·14-s + 0.796i·15-s + 0.387·16-s + 0.770·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.971246 - 0.826930i\)
\(L(\frac12)\) \(\approx\) \(0.971246 - 0.826930i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 + (-7.47 - 46.2i)T \)
good2 \( 1 + 1.32iT - 8T^{2} \)
5 \( 1 + 15.4iT - 125T^{2} \)
7 \( 1 + 7.96iT - 343T^{2} \)
11 \( 1 - 12.7iT - 1.33e3T^{2} \)
17 \( 1 - 54T + 4.91e3T^{2} \)
19 \( 1 - 84.5iT - 6.85e3T^{2} \)
23 \( 1 + 122.T + 1.21e4T^{2} \)
29 \( 1 - 140.T + 2.43e4T^{2} \)
31 \( 1 - 116. iT - 2.97e4T^{2} \)
37 \( 1 + 433. iT - 5.06e4T^{2} \)
41 \( 1 - 205. iT - 6.89e4T^{2} \)
43 \( 1 - 418.T + 7.95e4T^{2} \)
47 \( 1 - 485. iT - 1.03e5T^{2} \)
53 \( 1 + 674.T + 1.48e5T^{2} \)
59 \( 1 - 186. iT - 2.05e5T^{2} \)
61 \( 1 + 671.T + 2.26e5T^{2} \)
67 \( 1 + 14.0iT - 3.00e5T^{2} \)
71 \( 1 - 346. iT - 3.57e5T^{2} \)
73 \( 1 + 832. iT - 3.89e5T^{2} \)
79 \( 1 + 335.T + 4.93e5T^{2} \)
83 \( 1 + 568. iT - 5.71e5T^{2} \)
89 \( 1 + 236. iT - 7.04e5T^{2} \)
97 \( 1 - 1.27e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.11345733680589686685620302170, −14.17332675859921897832943550766, −12.51911692225590234100624093395, −12.11432783871335590511509138108, −10.70823296649040137584100108145, −9.469754281697169797448145605460, −7.68312977319347317698786087600, −6.02624074906807376649485194810, −4.26003509010226232746759957385, −1.38902586585077007976741595606, 2.86414057245313290155990252742, 5.70621185884193260495203017744, 6.70314269080960166989203312143, 7.951028238859600759792953353962, 10.22574701848744742654627773305, 11.10775630827462632357549398066, 12.13657433529510165022401205615, 13.95234822766228455415274633300, 15.16603617918564407113875048719, 15.75401636688819884417627999456

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