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Properties

Label 2-39-39.8-c3-0-10
Degree $2$
Conductor $39$
Sign $0.222 + 0.974i$
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  + (2.26 − 2.26i)2-s + (2.15 − 4.72i)3-s − 2.24i·4-s + (−0.350 + 0.350i)5-s + (−5.82 − 15.5i)6-s + (−7.24 + 7.24i)7-s + (13.0 + 13.0i)8-s + (−17.7 − 20.3i)9-s + 1.58i·10-s + (0.0601 + 0.0601i)11-s + (−10.6 − 4.83i)12-s + (38.0 + 27.4i)13-s + 32.7i·14-s + (0.902 + 2.41i)15-s + 76.9·16-s − 58.1·17-s + ⋯
L(s)  = 1  + (0.800 − 0.800i)2-s + (0.414 − 0.910i)3-s − 0.280i·4-s + (−0.0313 + 0.0313i)5-s + (−0.396 − 1.05i)6-s + (−0.390 + 0.390i)7-s + (0.575 + 0.575i)8-s + (−0.656 − 0.754i)9-s + 0.0501i·10-s + (0.00164 + 0.00164i)11-s + (−0.255 − 0.116i)12-s + (0.810 + 0.585i)13-s + 0.625i·14-s + (0.0155 + 0.0415i)15-s + 1.20·16-s − 0.829·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.222 + 0.974i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.222 + 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56814 - 1.25062i\)
\(L(\frac12)\) \(\approx\) \(1.56814 - 1.25062i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.15 + 4.72i)T \)
13 \( 1 + (-38.0 - 27.4i)T \)
good2 \( 1 + (-2.26 + 2.26i)T - 8iT^{2} \)
5 \( 1 + (0.350 - 0.350i)T - 125iT^{2} \)
7 \( 1 + (7.24 - 7.24i)T - 343iT^{2} \)
11 \( 1 + (-0.0601 - 0.0601i)T + 1.33e3iT^{2} \)
17 \( 1 + 58.1T + 4.91e3T^{2} \)
19 \( 1 + (-20.4 - 20.4i)T + 6.85e3iT^{2} \)
23 \( 1 + 93.7T + 1.21e4T^{2} \)
29 \( 1 + 189. iT - 2.43e4T^{2} \)
31 \( 1 + (173. + 173. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-247. + 247. i)T - 5.06e4iT^{2} \)
41 \( 1 + (132. - 132. i)T - 6.89e4iT^{2} \)
43 \( 1 + 277. iT - 7.95e4T^{2} \)
47 \( 1 + (219. + 219. i)T + 1.03e5iT^{2} \)
53 \( 1 - 582. iT - 1.48e5T^{2} \)
59 \( 1 + (-396. - 396. i)T + 2.05e5iT^{2} \)
61 \( 1 + 244.T + 2.26e5T^{2} \)
67 \( 1 + (44.4 + 44.4i)T + 3.00e5iT^{2} \)
71 \( 1 + (-454. + 454. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-430. + 430. i)T - 3.89e5iT^{2} \)
79 \( 1 - 637.T + 4.93e5T^{2} \)
83 \( 1 + (-868. + 868. i)T - 5.71e5iT^{2} \)
89 \( 1 + (-477. - 477. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-168. - 168. i)T + 9.12e5iT^{2} \)
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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

−15.10230597773039397798299544313, −13.78060615148381939662890602246, −13.16621764642086336542244007871, −12.06971945642531555138636962634, −11.16659292127733607904013897872, −9.168401890941835968927049394170, −7.72658293863460305996687276050, −6.02581337777403706066716986851, −3.79405283306520744475946368059, −2.14759921789959466389099374836, 3.66008051823978854199126261805, 5.04384332208060377508589464832, 6.54571856969306099568626614309, 8.263124131207175882783714836800, 9.850648038333993144842425725839, 10.93343381906865072350415867785, 12.97314035186937959720947441643, 13.92235543648436100421556566933, 14.86225872647856089519144130955, 15.94621339444185346981179830269

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