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Properties

Label 2-39-13.4-c3-0-1
Degree $2$
Conductor $39$
Sign $0.880 + 0.473i$
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  + (−4.64 − 2.68i)2-s + (−1.5 + 2.59i)3-s + (10.3 + 17.9i)4-s − 2.69i·5-s + (13.9 − 8.04i)6-s + (13.1 − 7.60i)7-s − 68.5i·8-s + (−4.5 − 7.79i)9-s + (−7.23 + 12.5i)10-s + (57.9 + 33.4i)11-s − 62.3·12-s + (46.8 − 0.818i)13-s − 81.5·14-s + (7.00 + 4.04i)15-s + (−100. + 174. i)16-s + (2.08 + 3.60i)17-s + ⋯
L(s)  = 1  + (−1.64 − 0.948i)2-s + (−0.288 + 0.499i)3-s + (1.29 + 2.24i)4-s − 0.241i·5-s + (0.948 − 0.547i)6-s + (0.710 − 0.410i)7-s − 3.03i·8-s + (−0.166 − 0.288i)9-s + (−0.228 + 0.396i)10-s + (1.58 + 0.916i)11-s − 1.49·12-s + (0.999 − 0.0174i)13-s − 1.55·14-s + (0.120 + 0.0696i)15-s + (−1.57 + 2.72i)16-s + (0.0297 + 0.0514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.880 + 0.473i$
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: \((2,\ 39,\ (\ :3/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.622651 - 0.156771i\)
\(L(\frac12)\) \(\approx\) \(0.622651 - 0.156771i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.8 + 0.818i)T \)
good2 \( 1 + (4.64 + 2.68i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 2.69iT - 125T^{2} \)
7 \( 1 + (-13.1 + 7.60i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-57.9 - 33.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-2.08 - 3.60i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (22.5 - 13.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-23.6 + 40.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (128. - 222. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 206. iT - 2.97e4T^{2} \)
37 \( 1 + (152. + 87.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-135. - 78.2i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-25.9 - 45.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 354. iT - 1.03e5T^{2} \)
53 \( 1 + 10.4T + 1.48e5T^{2} \)
59 \( 1 + (-385. + 222. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (59.8 + 103. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (19.4 + 11.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (246. - 142. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 740. iT - 3.89e5T^{2} \)
79 \( 1 + 547.T + 4.93e5T^{2} \)
83 \( 1 + 603. iT - 5.71e5T^{2} \)
89 \( 1 + (186. + 107. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.25e3 - 723. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−16.32938567095872228717462259172, −14.72071545697392625585638939429, −12.63969395362258216400791199430, −11.49953778255321839603718467753, −10.71654669506449708703883670603, −9.449393873139207754410182829661, −8.546188157844403919905900378829, −6.96311020728326541393558082562, −3.96998858986611863447739763420, −1.38509889721573552728407001318, 1.29355724060580582795144123452, 5.85567658778609989408559342005, 6.85031535397446846567272208883, 8.308245967363241276094332370760, 9.110977150610393664981133532988, 10.85097618622603021934244678939, 11.61938493020730607551871239674, 13.98223669523760445367949486098, 14.98028048397635743090467284620, 16.21822799763563232784896748378

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