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Properties

Label 2-39-13.3-c3-0-3
Degree $2$
Conductor $39$
Sign $0.448 + 0.893i$
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  + (−1.18 − 2.04i)2-s + (1.5 + 2.59i)3-s + (1.21 − 2.09i)4-s + 6.42·5-s + (3.54 − 6.13i)6-s + (14.7 − 25.5i)7-s − 24.6·8-s + (−4.5 + 7.79i)9-s + (−7.58 − 13.1i)10-s + (0.312 + 0.541i)11-s + 7.26·12-s + (44.3 + 15.0i)13-s − 69.6·14-s + (9.63 + 16.6i)15-s + (19.3 + 33.5i)16-s + (−43.8 + 75.9i)17-s + ⋯
L(s)  = 1  + (−0.417 − 0.723i)2-s + (0.288 + 0.499i)3-s + (0.151 − 0.262i)4-s + 0.574·5-s + (0.241 − 0.417i)6-s + (0.796 − 1.37i)7-s − 1.08·8-s + (−0.166 + 0.288i)9-s + (−0.239 − 0.415i)10-s + (0.00856 + 0.0148i)11-s + 0.174·12-s + (0.947 + 0.320i)13-s − 1.32·14-s + (0.165 + 0.287i)15-s + (0.302 + 0.524i)16-s + (−0.625 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11567 - 0.688304i\)
\(L(\frac12)\) \(\approx\) \(1.11567 - 0.688304i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-44.3 - 15.0i)T \)
good2 \( 1 + (1.18 + 2.04i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 6.42T + 125T^{2} \)
7 \( 1 + (-14.7 + 25.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-0.312 - 0.541i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (43.8 - 75.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (41.4 - 71.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-37.3 - 64.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (113. + 196. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 173.T + 2.97e4T^{2} \)
37 \( 1 + (56.0 + 97.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-133. - 231. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (191. - 332. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 337.T + 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 + (-264. + 458. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (101. - 176. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (60.7 + 105. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-330. + 572. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 167.T + 3.89e5T^{2} \)
79 \( 1 + 101.T + 4.93e5T^{2} \)
83 \( 1 + 506.T + 5.71e5T^{2} \)
89 \( 1 + (701. + 1.21e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (951. - 1.64e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.42283305656695887613460986760, −14.32007186465733756116126679739, −13.30646261340609230449700991918, −11.36136997089097879785458559926, −10.60313848697579605649656403404, −9.657348794721982074006389664159, −8.171265630515781316834183943453, −6.14213184647518292967682321681, −4.02843953520148453423431929321, −1.64312041841891420260122320313, 2.49694898234205079944303046568, 5.58419892026847278499952027737, 6.90320700198028839329793105671, 8.449622337143429214485049364845, 9.045702743281782401859976843410, 11.28096613217155889423985999627, 12.39419635037775347030175497319, 13.70610997293742277043317230559, 15.07850975111725124749029959859, 15.80116707159507816429434743813

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