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Properties

Label 2-39-13.4-c3-0-0
Degree $2$
Conductor $39$
Sign $-0.525 - 0.851i$
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  + (−1.76 − 1.02i)2-s + (−1.5 + 2.59i)3-s + (−1.91 − 3.31i)4-s + 12.0i·5-s + (5.30 − 3.06i)6-s + (−25.7 + 14.8i)7-s + 24.1i·8-s + (−4.5 − 7.79i)9-s + (12.3 − 21.3i)10-s + (24.3 + 14.0i)11-s + 11.4·12-s + (−40.9 − 22.7i)13-s + 60.7·14-s + (−31.3 − 18.1i)15-s + (9.35 − 16.1i)16-s + (−25.3 − 43.8i)17-s + ⋯
L(s)  = 1  + (−0.625 − 0.361i)2-s + (−0.288 + 0.499i)3-s + (−0.239 − 0.414i)4-s + 1.08i·5-s + (0.361 − 0.208i)6-s + (−1.39 + 0.802i)7-s + 1.06i·8-s + (−0.166 − 0.288i)9-s + (0.390 − 0.675i)10-s + (0.666 + 0.384i)11-s + 0.276·12-s + (−0.874 − 0.485i)13-s + 1.15·14-s + (−0.540 − 0.311i)15-s + (0.146 − 0.253i)16-s + (−0.361 − 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.525 - 0.851i$
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: \((2,\ 39,\ (\ :3/2),\ -0.525 - 0.851i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.205064 + 0.367485i\)
\(L(\frac12)\) \(\approx\) \(0.205064 + 0.367485i\)
\(L(\frac{5}{2})\) 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 \( 1 + (40.9 + 22.7i)T \)
good2 \( 1 + (1.76 + 1.02i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 12.0iT - 125T^{2} \)
7 \( 1 + (25.7 - 14.8i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-24.3 - 14.0i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (25.3 + 43.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-91.0 + 52.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (80.2 - 139. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (70.0 - 121. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 223. iT - 2.97e4T^{2} \)
37 \( 1 + (197. + 114. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-256. - 147. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-96.0 - 166. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 36.9iT - 1.03e5T^{2} \)
53 \( 1 - 149.T + 1.48e5T^{2} \)
59 \( 1 + (380. - 219. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (143. + 247. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-465. - 268. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-88.9 + 51.3i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 75.5iT - 3.89e5T^{2} \)
79 \( 1 - 17.5T + 4.93e5T^{2} \)
83 \( 1 - 1.46e3iT - 5.71e5T^{2} \)
89 \( 1 + (290. + 167. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (648. - 374. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−16.03317019471565696270976830159, −15.09169724361594859957798558197, −14.01291221112426582029265757375, −12.22757587531071911565928937877, −11.01916085426739010892371408319, −9.784188708236655230519418990179, −9.297815221422881554592351839050, −6.96451625399292103968388608311, −5.46765559899408091864602775538, −2.96287965405700972635709823728, 0.44252120157333791241515461499, 4.04906305147848961248537148533, 6.34583066508708487404185016338, 7.60353479332672980997296528959, 8.988165782113877105596535018450, 9.975388262854962172674348896250, 12.11138515289537244360122868811, 12.84746585774451996722514172268, 13.86002389789380273452267299145, 16.01252664609319720183808427469

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