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Properties

Label 2-39-39.8-c3-0-9
Degree $2$
Conductor $39$
Sign $0.492 + 0.870i$
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  + (3.62 − 3.62i)2-s + (4.19 + 3.07i)3-s − 18.2i·4-s + (−10.8 + 10.8i)5-s + (26.3 − 4.05i)6-s + (−5.52 + 5.52i)7-s + (−37.1 − 37.1i)8-s + (8.11 + 25.7i)9-s + 78.9i·10-s + (−25.6 − 25.6i)11-s + (56.1 − 76.5i)12-s + (6.01 − 46.4i)13-s + 40.0i·14-s + (−79.1 + 12.1i)15-s − 123.·16-s + 56.1·17-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)2-s + (0.806 + 0.591i)3-s − 2.28i·4-s + (−0.974 + 0.974i)5-s + (1.79 − 0.275i)6-s + (−0.298 + 0.298i)7-s + (−1.64 − 1.64i)8-s + (0.300 + 0.953i)9-s + 2.49i·10-s + (−0.702 − 0.702i)11-s + (1.35 − 1.84i)12-s + (0.128 − 0.991i)13-s + 0.764i·14-s + (−1.36 + 0.209i)15-s − 1.92·16-s + 0.800·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.492 + 0.870i$
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: \((2,\ 39,\ (\ :3/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.10058 - 1.22430i\)
\(L(\frac12)\) \(\approx\) \(2.10058 - 1.22430i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.19 - 3.07i)T \)
13 \( 1 + (-6.01 + 46.4i)T \)
good2 \( 1 + (-3.62 + 3.62i)T - 8iT^{2} \)
5 \( 1 + (10.8 - 10.8i)T - 125iT^{2} \)
7 \( 1 + (5.52 - 5.52i)T - 343iT^{2} \)
11 \( 1 + (25.6 + 25.6i)T + 1.33e3iT^{2} \)
17 \( 1 - 56.1T + 4.91e3T^{2} \)
19 \( 1 + (-72.4 - 72.4i)T + 6.85e3iT^{2} \)
23 \( 1 - 8.33T + 1.21e4T^{2} \)
29 \( 1 + 125. iT - 2.43e4T^{2} \)
31 \( 1 + (89.8 + 89.8i)T + 2.97e4iT^{2} \)
37 \( 1 + (274. - 274. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-44.2 + 44.2i)T - 6.89e4iT^{2} \)
43 \( 1 + 100. iT - 7.95e4T^{2} \)
47 \( 1 + (82.5 + 82.5i)T + 1.03e5iT^{2} \)
53 \( 1 - 31.7iT - 1.48e5T^{2} \)
59 \( 1 + (150. + 150. i)T + 2.05e5iT^{2} \)
61 \( 1 - 383.T + 2.26e5T^{2} \)
67 \( 1 + (-227. - 227. i)T + 3.00e5iT^{2} \)
71 \( 1 + (298. - 298. i)T - 3.57e5iT^{2} \)
73 \( 1 + (560. - 560. i)T - 3.89e5iT^{2} \)
79 \( 1 - 805.T + 4.93e5T^{2} \)
83 \( 1 + (-828. + 828. i)T - 5.71e5iT^{2} \)
89 \( 1 + (-373. - 373. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-254. - 254. i)T + 9.12e5iT^{2} \)
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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

−15.15696045415129339068347788587, −14.26403451127618346573421191578, −13.23606621575597194743872913307, −11.92117296822993705857588975826, −10.79248068776417948754898776555, −9.994057183323091718507590892008, −7.904732551454848097579268914084, −5.49111928584571590962600097118, −3.62937629354675992389210432448, −2.94464821927610250646880297757, 3.61064213801246174858404486148, 4.96609016996671962286291788762, 6.98396912320493084862833288558, 7.75369693857728124037903538343, 8.998051992865978354474128649349, 12.02978498408137258639160124853, 12.73287941738125076953446193378, 13.69250739987486240820559439171, 14.72623643634888480164981777137, 15.82770788616347921032288997338

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