↑

Properties

Label 2-39-13.4-c3-0-7
Degree $2$
Conductor $39$
Sign $-0.998 - 0.0573i$
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  + (−3.57 − 2.06i)2-s + (1.5 − 2.59i)3-s + (4.50 + 7.79i)4-s − 13.4i·5-s + (−10.7 + 6.18i)6-s + (−27.2 + 15.7i)7-s − 4.12i·8-s + (−4.5 − 7.79i)9-s + (−27.7 + 47.9i)10-s + (−35.0 − 20.2i)11-s + 27.0·12-s + (42.1 − 20.5i)13-s + 129.·14-s + (−34.9 − 20.1i)15-s + (27.4 − 47.6i)16-s + (−21.5 − 37.3i)17-s + ⋯
L(s)  = 1  + (−1.26 − 0.728i)2-s + (0.288 − 0.499i)3-s + (0.562 + 0.974i)4-s − 1.20i·5-s + (−0.728 + 0.420i)6-s + (−1.46 + 0.848i)7-s − 0.182i·8-s + (−0.166 − 0.288i)9-s + (−0.876 + 1.51i)10-s + (−0.961 − 0.555i)11-s + 0.649·12-s + (0.898 − 0.438i)13-s + 2.47·14-s + (−0.601 − 0.347i)15-s + (0.429 − 0.744i)16-s + (−0.307 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0573i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.998 - 0.0573i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.998 - 0.0573i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0128761 + 0.448686i\)
\(L(\frac12)\) \(\approx\) \(0.0128761 + 0.448686i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-42.1 + 20.5i)T \)
good2 \( 1 + (3.57 + 2.06i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 13.4iT - 125T^{2} \)
7 \( 1 + (27.2 - 15.7i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (35.0 + 20.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (21.5 + 37.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-23.3 + 13.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (9.50 - 16.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-77.0 + 133. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 308. iT - 2.97e4T^{2} \)
37 \( 1 + (37.6 + 21.7i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-41.4 - 23.9i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-171. - 296. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 133. iT - 1.03e5T^{2} \)
53 \( 1 + 438.T + 1.48e5T^{2} \)
59 \( 1 + (-511. + 295. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-270. - 468. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (199. + 115. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (389. - 224. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 389. iT - 3.89e5T^{2} \)
79 \( 1 + 897.T + 4.93e5T^{2} \)
83 \( 1 - 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 + (-801. - 462. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.35e3 + 780. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−15.71790626734812680290644657218, −13.37555518851625846592237256598, −12.68859128094707538602600611360, −11.43854107125190542435127332301, −9.791034634738723748287434986914, −8.947753212563347967369678117224, −8.020645559262573432685455299044, −5.82520214043378509770827240192, −2.75537342279292489765590682191, −0.53377136543492187555968545806, 3.44301783024545184853557167260, 6.47291460446494201590771646096, 7.29763802444551228660047312993, 8.866562938068789930750052983620, 10.27811045048479165909652027449, 10.50422709266487798097014986255, 13.01016566552357720410455458100, 14.35195807109856558262729279908, 15.77122176988081621379118791478, 16.09003191765808513741895079112

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