Properties

Label 2-39-39.2-c3-0-5
Degree $2$
Conductor $39$
Sign $0.982 - 0.188i$
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  + (0.532 − 1.98i)2-s + (−0.220 + 5.19i)3-s + (3.26 + 1.88i)4-s + (4.41 + 4.41i)5-s + (10.1 + 3.20i)6-s + (3.85 − 1.03i)7-s + (17.1 − 17.1i)8-s + (−26.9 − 2.29i)9-s + (11.1 − 6.41i)10-s + (5.64 + 1.51i)11-s + (−10.5 + 16.5i)12-s + (−11.9 − 45.3i)13-s − 8.21i·14-s + (−23.8 + 21.9i)15-s + (−9.78 − 16.9i)16-s + (−53.6 + 92.8i)17-s + ⋯
L(s)  = 1  + (0.188 − 0.702i)2-s + (−0.0424 + 0.999i)3-s + (0.408 + 0.235i)4-s + (0.394 + 0.394i)5-s + (0.693 + 0.217i)6-s + (0.208 − 0.0558i)7-s + (0.756 − 0.756i)8-s + (−0.996 − 0.0848i)9-s + (0.351 − 0.203i)10-s + (0.154 + 0.0414i)11-s + (−0.252 + 0.398i)12-s + (−0.254 − 0.967i)13-s − 0.156i·14-s + (−0.411 + 0.377i)15-s + (−0.152 − 0.264i)16-s + (−0.764 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.982 - 0.188i$
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: \((2,\ 39,\ (\ :3/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.59920 + 0.151816i\)
\(L(\frac12)\) \(\approx\) \(1.59920 + 0.151816i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.220 - 5.19i)T \)
13 \( 1 + (11.9 + 45.3i)T \)
good2 \( 1 + (-0.532 + 1.98i)T + (-6.92 - 4i)T^{2} \)
5 \( 1 + (-4.41 - 4.41i)T + 125iT^{2} \)
7 \( 1 + (-3.85 + 1.03i)T + (297. - 171.5i)T^{2} \)
11 \( 1 + (-5.64 - 1.51i)T + (1.15e3 + 665.5i)T^{2} \)
17 \( 1 + (53.6 - 92.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (15.7 + 58.6i)T + (-5.94e3 + 3.42e3i)T^{2} \)
23 \( 1 + (56.5 + 97.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-118. + 68.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (10.7 - 10.7i)T - 2.97e4iT^{2} \)
37 \( 1 + (55.7 - 208. i)T + (-4.38e4 - 2.53e4i)T^{2} \)
41 \( 1 + (83.3 - 311. i)T + (-5.96e4 - 3.44e4i)T^{2} \)
43 \( 1 + (-62.0 - 35.7i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-258. + 258. i)T - 1.03e5iT^{2} \)
53 \( 1 + 216. iT - 1.48e5T^{2} \)
59 \( 1 + (-166. - 621. i)T + (-1.77e5 + 1.02e5i)T^{2} \)
61 \( 1 + (371. - 644. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (335. + 90.0i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + (347. - 93.1i)T + (3.09e5 - 1.78e5i)T^{2} \)
73 \( 1 + (-761. - 761. i)T + 3.89e5iT^{2} \)
79 \( 1 - 670.T + 4.93e5T^{2} \)
83 \( 1 + (1.03e3 + 1.03e3i)T + 5.71e5iT^{2} \)
89 \( 1 + (-841. - 225. i)T + (6.10e5 + 3.52e5i)T^{2} \)
97 \( 1 + (-9.67 - 36.1i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−15.66655112299543981858602555437, −14.75502444618454132599488701027, −13.29631398694305243340997127490, −11.95574648386834281416206005300, −10.70660770169757169854301893890, −10.15669774358302499471312385255, −8.346364341307225988097854445849, −6.35055688945708062474213907783, −4.35276112940722624457406592367, −2.69869947798531341196252478966, 1.89449762321366512715333234970, 5.24267197064491767173598717110, 6.55207455131161802699024625140, 7.60786528663877428787944465303, 9.153228513307208523153129960695, 11.12750315344822128077514856371, 12.17617331468862725262073578195, 13.71224039184808402348420684646, 14.28324430766890462118780718054, 15.78706061495352454723535220487

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