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Properties

Label 2-39-13.9-c3-0-3
Degree $2$
Conductor $39$
Sign $0.756 + 0.653i$
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  + (0.733 − 1.26i)2-s + (1.5 − 2.59i)3-s + (2.92 + 5.06i)4-s + 9.85·5-s + (−2.19 − 3.80i)6-s + (−14.9 − 25.9i)7-s + 20.3·8-s + (−4.5 − 7.79i)9-s + (7.22 − 12.5i)10-s + (−23.4 + 40.6i)11-s + 17.5·12-s + (3.71 + 46.7i)13-s − 43.8·14-s + (14.7 − 25.5i)15-s + (−8.51 + 14.7i)16-s + (24.1 + 41.7i)17-s + ⋯
L(s)  = 1  + (0.259 − 0.448i)2-s + (0.288 − 0.499i)3-s + (0.365 + 0.633i)4-s + 0.881·5-s + (−0.149 − 0.259i)6-s + (−0.808 − 1.40i)7-s + 0.897·8-s + (−0.166 − 0.288i)9-s + (0.228 − 0.395i)10-s + (−0.643 + 1.11i)11-s + 0.422·12-s + (0.0791 + 0.996i)13-s − 0.837·14-s + (0.254 − 0.440i)15-s + (−0.133 + 0.230i)16-s + (0.344 + 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.756 + 0.653i$
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: \((2,\ 39,\ (\ :3/2),\ 0.756 + 0.653i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63346 - 0.607916i\)
\(L(\frac12)\) \(\approx\) \(1.63346 - 0.607916i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.71 - 46.7i)T \)
good2 \( 1 + (-0.733 + 1.26i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 9.85T + 125T^{2} \)
7 \( 1 + (14.9 + 25.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (23.4 - 40.6i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-24.1 - 41.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (60.1 + 104. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (65.3 - 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-97.4 + 168. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 32.0T + 2.97e4T^{2} \)
37 \( 1 + (-16.2 + 28.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-120. + 209. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (48.2 + 83.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 539.T + 1.03e5T^{2} \)
53 \( 1 + 152.T + 1.48e5T^{2} \)
59 \( 1 + (163. + 283. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-49.2 - 85.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-220. + 382. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (172. + 298. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 773.T + 3.89e5T^{2} \)
79 \( 1 + 150.T + 4.93e5T^{2} \)
83 \( 1 - 337.T + 5.71e5T^{2} \)
89 \( 1 + (84.9 - 147. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (107. + 185. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.67736077033707474128760108534, −13.80919761535104354066093721943, −13.33339788874626756337131375301, −12.30207835737321879590331387718, −10.71642367552750480454395771419, −9.600229654573189355684491275446, −7.60753050496421013823155461745, −6.63336306967672976167045086435, −4.06750585157809879574701369756, −2.15084903981435085023294083703, 2.68782520551745971186160421225, 5.50974473158194375459090705159, 6.08950498372324192712018022582, 8.360658362736112769712613443083, 9.770839008433808703492011572315, 10.66994139841973670712233820336, 12.52479132612473023291258395486, 13.80636212503195937970066954749, 14.80724029664174434335981232576, 15.85589622586651317931963505122

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