↑

Properties

Label 2-39-13.5-c2-0-2
Degree $2$
Conductor $39$
Sign $0.766 + 0.642i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 1.67i)2-s + 1.73·3-s − 1.62i·4-s + (−4.84 + 4.84i)5-s + (2.90 − 2.90i)6-s + (−8.89 − 8.89i)7-s + (3.98 + 3.98i)8-s + 2.99·9-s + 16.2i·10-s + (−0.161 − 0.161i)11-s − 2.80i·12-s + (10.8 − 7.11i)13-s − 29.8·14-s + (−8.39 + 8.39i)15-s + 19.8·16-s + 15.8i·17-s + ⋯
L(s)  = 1  + (0.838 − 0.838i)2-s + 0.577·3-s − 0.405i·4-s + (−0.969 + 0.969i)5-s + (0.483 − 0.483i)6-s + (−1.27 − 1.27i)7-s + (0.498 + 0.498i)8-s + 0.333·9-s + 1.62i·10-s + (−0.0146 − 0.0146i)11-s − 0.233i·12-s + (0.837 − 0.547i)13-s − 2.12·14-s + (−0.559 + 0.559i)15-s + 1.24·16-s + 0.933i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42981 - 0.520300i\)
\(L(\frac12)\) \(\approx\) \(1.42981 - 0.520300i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
13 \( 1 + (-10.8 + 7.11i)T \)
good2 \( 1 + (-1.67 + 1.67i)T - 4iT^{2} \)
5 \( 1 + (4.84 - 4.84i)T - 25iT^{2} \)
7 \( 1 + (8.89 + 8.89i)T + 49iT^{2} \)
11 \( 1 + (0.161 + 0.161i)T + 121iT^{2} \)
17 \( 1 - 15.8iT - 289T^{2} \)
19 \( 1 + (0.414 - 0.414i)T - 361iT^{2} \)
23 \( 1 + 19.1iT - 529T^{2} \)
29 \( 1 - 2.28T + 841T^{2} \)
31 \( 1 + (9.32 - 9.32i)T - 961iT^{2} \)
37 \( 1 + (29.1 + 29.1i)T + 1.36e3iT^{2} \)
41 \( 1 + (47.0 - 47.0i)T - 1.68e3iT^{2} \)
43 \( 1 - 0.706iT - 1.84e3T^{2} \)
47 \( 1 + (7.40 + 7.40i)T + 2.20e3iT^{2} \)
53 \( 1 - 14.5T + 2.80e3T^{2} \)
59 \( 1 + (-22.0 - 22.0i)T + 3.48e3iT^{2} \)
61 \( 1 + 9.71T + 3.72e3T^{2} \)
67 \( 1 + (-86.4 + 86.4i)T - 4.48e3iT^{2} \)
71 \( 1 + (-59.4 + 59.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (-67.7 - 67.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 5.39T + 6.24e3T^{2} \)
83 \( 1 + (28.3 - 28.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (109. + 109. i)T + 7.92e3iT^{2} \)
97 \( 1 + (-59.5 + 59.5i)T - 9.40e3iT^{2} \)
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

−15.61389079545728543226657440261, −14.40784064311627945899404070704, −13.39144165319951361103204429613, −12.54379970015502886866822654420, −10.97704338105091400579015040624, −10.29493546624707702096715298507, −8.071766279117858333395726690028, −6.76434916244533683253249822730, −3.90536568090514596071813523202, −3.26048908293638691594828324675, 3.69011759400452996374586321219, 5.32791478886178221715932994748, 6.83686424592796458619977828708, 8.454459395486212104410029987896, 9.533426824151444083129048970679, 11.88317380408179468896660132885, 12.86145325500185186317538375285, 13.80102290026008775350733173817, 15.37894169769401052532592138170, 15.75669671574153378254106433931

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