Properties

Label 2-39-13.10-c3-0-0
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

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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: $\chi_{39} (10, \cdot )$
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 \( 1 + (-1.35e3 - 780. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

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

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