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Properties

Label 2-39-13.2-c2-0-2
Degree $2$
Conductor $39$
Sign $0.919 - 0.394i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 + 0.463i)2-s + (0.866 + 1.5i)3-s + (−0.691 − 0.399i)4-s + (0.707 − 0.707i)5-s + (0.802 + 2.99i)6-s + (−2.02 + 0.543i)7-s + (−6.07 − 6.07i)8-s + (−1.5 + 2.59i)9-s + (1.55 − 0.895i)10-s + (2.74 − 10.2i)11-s − 1.38i·12-s + (−1.20 + 12.9i)13-s − 3.75·14-s + (1.67 + 0.448i)15-s + (−6.08 − 10.5i)16-s + (−8.98 − 5.18i)17-s + ⋯
L(s)  = 1  + (0.864 + 0.231i)2-s + (0.288 + 0.5i)3-s + (−0.172 − 0.0998i)4-s + (0.141 − 0.141i)5-s + (0.133 + 0.498i)6-s + (−0.289 + 0.0775i)7-s + (−0.758 − 0.758i)8-s + (−0.166 + 0.288i)9-s + (0.155 − 0.0895i)10-s + (0.249 − 0.931i)11-s − 0.115i·12-s + (−0.0925 + 0.995i)13-s − 0.268·14-s + (0.111 + 0.0299i)15-s + (−0.380 − 0.658i)16-s + (−0.528 − 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.919 - 0.394i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ 0.919 - 0.394i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45493 + 0.298846i\)
\(L(\frac12)\) \(\approx\) \(1.45493 + 0.298846i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
13 \( 1 + (1.20 - 12.9i)T \)
good2 \( 1 + (-1.72 - 0.463i)T + (3.46 + 2i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T - 25iT^{2} \)
7 \( 1 + (2.02 - 0.543i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-2.74 + 10.2i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (8.98 + 5.18i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.44 - 20.3i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-21.6 + 12.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-12.1 - 20.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-18.1 + 18.1i)T - 961iT^{2} \)
37 \( 1 + (-7.82 + 29.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-23.9 - 6.43i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (72.3 + 41.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (45.3 + 45.3i)T + 2.20e3iT^{2} \)
53 \( 1 + 65.1T + 2.80e3T^{2} \)
59 \( 1 + (-74.5 + 19.9i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-13.2 + 23.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-74.9 - 20.0i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (27.4 + 102. i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-82.5 - 82.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 70.2T + 6.24e3T^{2} \)
83 \( 1 + (-9.18 + 9.18i)T - 6.88e3iT^{2} \)
89 \( 1 + (14.7 - 55.1i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (11.7 + 43.6i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−15.91353982482922048932140289316, −14.68613438070839078904209658002, −13.91711122540962545897873420322, −12.88355439851058989835281786426, −11.41112849165503445209337238060, −9.743355319017968805068222537288, −8.739533123004451261399699513109, −6.52827343986116804744454988478, −5.07246054828626169489668773777, −3.56769911569006616679307730878, 2.92169342248754909725123137733, 4.79039634929267519553093428192, 6.58690827256320507364615294741, 8.246624445899892142262824226560, 9.726751381064605614379747358780, 11.50741728523114003888227966359, 12.74746057620785631238627450389, 13.34625695598510133387796402438, 14.56176035026512926694462219145, 15.49775564830306010078881840343

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