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Properties

Label 2-39-13.9-c3-0-2
Degree $2$
Conductor $39$
Sign $0.859 - 0.511i$
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  + (0.5 − 0.866i)2-s + (−1.5 + 2.59i)3-s + (3.5 + 6.06i)4-s + 7·5-s + (1.5 + 2.59i)6-s + (5 + 8.66i)7-s + 15·8-s + (−4.5 − 7.79i)9-s + (3.5 − 6.06i)10-s + (11 − 19.0i)11-s − 21·12-s + (−45.5 − 11.2i)13-s + 10·14-s + (−10.5 + 18.1i)15-s + (−20.5 + 35.5i)16-s + (−18.5 − 32.0i)17-s + ⋯
L(s)  = 1  + (0.176 − 0.306i)2-s + (−0.288 + 0.499i)3-s + (0.437 + 0.757i)4-s + 0.626·5-s + (0.102 + 0.176i)6-s + (0.269 + 0.467i)7-s + 0.662·8-s + (−0.166 − 0.288i)9-s + (0.110 − 0.191i)10-s + (0.301 − 0.522i)11-s − 0.505·12-s + (−0.970 − 0.240i)13-s + 0.190·14-s + (−0.180 + 0.313i)15-s + (−0.320 + 0.554i)16-s + (−0.263 − 0.457i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.46054 + 0.401407i\)
\(L(\frac12)\) \(\approx\) \(1.46054 + 0.401407i\)
\(L(\frac{5}{2})\) 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 \( 1 + (45.5 + 11.2i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 7T + 125T^{2} \)
7 \( 1 + (-5 - 8.66i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-11 + 19.0i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (18.5 + 32.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (15 + 25.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-81 + 140. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-56.5 + 97.8i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 196T + 2.97e4T^{2} \)
37 \( 1 + (6.5 - 11.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (142.5 - 246. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-123 - 213. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 462T + 1.03e5T^{2} \)
53 \( 1 + 537T + 1.48e5T^{2} \)
59 \( 1 + (288 + 498. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-317.5 - 549. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (101 - 174. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-543 - 940. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 805T + 3.89e5T^{2} \)
79 \( 1 - 884T + 4.93e5T^{2} \)
83 \( 1 - 518T + 5.71e5T^{2} \)
89 \( 1 + (97 - 168. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-601 - 1.04e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.96335157320361747400818033815, −14.71065406476176879426368043299, −13.33110249511723568313264495032, −12.10452018916418862968736305427, −11.15827264771961786886357061358, −9.792001252726823010044286848286, −8.302137017093270503925324577102, −6.51087938510444781682605443790, −4.74113648188002584159266979691, −2.69121489734570904503322389083, 1.73942523488203418917004659384, 4.98122494267059200731253698901, 6.38530517900362351922164538031, 7.49195676063671437102792455199, 9.612213938687824173923577419352, 10.72958571480985644882449018400, 12.05486450592735556234436127103, 13.52441183341948312103113357095, 14.40501914791398660770265232756, 15.50660185736389799146505023790

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