Properties

Label 2-39-39.2-c3-0-3
Degree $2$
Conductor $39$
Sign $0.267 - 0.963i$
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  + (−0.908 + 3.39i)2-s + (2.68 − 4.44i)3-s + (−3.73 − 2.15i)4-s + (12.1 + 12.1i)5-s + (12.6 + 13.1i)6-s + (−3.81 + 1.02i)7-s + (−9.13 + 9.13i)8-s + (−12.5 − 23.8i)9-s + (−52.1 + 30.1i)10-s + (7.61 + 2.03i)11-s + (−19.6 + 10.8i)12-s + (44.8 − 13.7i)13-s − 13.8i·14-s + (86.6 − 21.3i)15-s + (−39.9 − 69.1i)16-s + (−17.2 + 29.8i)17-s + ⋯
L(s)  = 1  + (−0.321 + 1.19i)2-s + (0.516 − 0.856i)3-s + (−0.467 − 0.269i)4-s + (1.08 + 1.08i)5-s + (0.859 + 0.894i)6-s + (−0.206 + 0.0552i)7-s + (−0.403 + 0.403i)8-s + (−0.465 − 0.885i)9-s + (−1.65 + 0.952i)10-s + (0.208 + 0.0559i)11-s + (−0.472 + 0.260i)12-s + (0.955 − 0.293i)13-s − 0.264i·14-s + (1.49 − 0.368i)15-s + (−0.624 − 1.08i)16-s + (−0.246 + 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.267 - 0.963i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.267 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14011 + 0.866356i\)
\(L(\frac12)\) \(\approx\) \(1.14011 + 0.866356i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.68 + 4.44i)T \)
13 \( 1 + (-44.8 + 13.7i)T \)
good2 \( 1 + (0.908 - 3.39i)T + (-6.92 - 4i)T^{2} \)
5 \( 1 + (-12.1 - 12.1i)T + 125iT^{2} \)
7 \( 1 + (3.81 - 1.02i)T + (297. - 171.5i)T^{2} \)
11 \( 1 + (-7.61 - 2.03i)T + (1.15e3 + 665.5i)T^{2} \)
17 \( 1 + (17.2 - 29.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (26.2 + 97.9i)T + (-5.94e3 + 3.42e3i)T^{2} \)
23 \( 1 + (98.5 + 170. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-60.6 + 35.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (70.3 - 70.3i)T - 2.97e4iT^{2} \)
37 \( 1 + (-38.6 + 144. i)T + (-4.38e4 - 2.53e4i)T^{2} \)
41 \( 1 + (82.0 - 306. i)T + (-5.96e4 - 3.44e4i)T^{2} \)
43 \( 1 + (-410. - 237. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (150. - 150. i)T - 1.03e5iT^{2} \)
53 \( 1 - 401. iT - 1.48e5T^{2} \)
59 \( 1 + (146. + 545. i)T + (-1.77e5 + 1.02e5i)T^{2} \)
61 \( 1 + (-272. + 471. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-490. - 131. i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + (138. - 37.0i)T + (3.09e5 - 1.78e5i)T^{2} \)
73 \( 1 + (-74.5 - 74.5i)T + 3.89e5iT^{2} \)
79 \( 1 + 341.T + 4.93e5T^{2} \)
83 \( 1 + (-465. - 465. i)T + 5.71e5iT^{2} \)
89 \( 1 + (514. + 137. i)T + (6.10e5 + 3.52e5i)T^{2} \)
97 \( 1 + (-437. - 1.63e3i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−15.89031011031911433002306373158, −14.67560215968691626165186745636, −14.06050097784228173169020181827, −12.84603891168796184633416077542, −11.00317960943029454618146508331, −9.317144388850068006859453345184, −8.105107300794892507529793668781, −6.61100325351505645773137238541, −6.20709146787950855420480842993, −2.60282747841248606792642821834, 1.79835368688529991561865824219, 3.81806030469183697845215336569, 5.74472858046288672719141284757, 8.643014529656761444765432112378, 9.502458569252892394996607388972, 10.31059307452038702683674133724, 11.69211449527560494409547998235, 13.08950946820314946216958575075, 13.98187412347473574569747538471, 15.69662348099703045189056079903

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