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Properties

Label 2-39-39.20-c5-0-16
Degree $2$
Conductor $39$
Sign $0.190 + 0.981i$
Analytic cond. $6.25496$
Root an. cond. $2.50099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.79 − 13.5i)3-s + (27.7 − 16i)4-s + (−41.1 − 11.0i)7-s + (−121.5 − 210. i)9-s − 498. i·12-s + (470. − 387.5i)13-s + (511. − 886. i)16-s + (−250. + 936. i)19-s + (−470. + 470. i)21-s + 3.12e3i·25-s − 3.78e3·27-s + (−1.31e3 + 353. i)28-s + (6.53e3 + 6.53e3i)31-s + (−6.73e3 − 3.88e3i)36-s + (3.36e3 + 1.25e4i)37-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.866 − 0.5i)4-s + (−0.317 − 0.0851i)7-s + (−0.5 − 0.866i)9-s − 0.999i·12-s + (0.771 − 0.635i)13-s + (0.499 − 0.866i)16-s + (−0.159 + 0.594i)19-s + (−0.232 + 0.232i)21-s + i·25-s − 1.00·27-s + (−0.317 + 0.0851i)28-s + (1.22 + 1.22i)31-s + (−0.866 − 0.5i)36-s + (0.404 + 1.51i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.190 + 0.981i$
Analytic conductor: \(6.25496\)
Root analytic conductor: \(2.50099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :5/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.61903 - 1.33540i\)
\(L(\frac12)\) \(\approx\) \(1.61903 - 1.33540i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.79 + 13.5i)T \)
13 \( 1 + (-470. + 387.5i)T \)
good2 \( 1 + (-27.7 + 16i)T^{2} \)
5 \( 1 - 3.12e3iT^{2} \)
7 \( 1 + (41.1 + 11.0i)T + (1.45e4 + 8.40e3i)T^{2} \)
11 \( 1 + (1.39e5 - 8.05e4i)T^{2} \)
17 \( 1 + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (250. - 936. i)T + (-2.14e6 - 1.23e6i)T^{2} \)
23 \( 1 + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-6.53e3 - 6.53e3i)T + 2.86e7iT^{2} \)
37 \( 1 + (-3.36e3 - 1.25e4i)T + (-6.00e7 + 3.46e7i)T^{2} \)
41 \( 1 + (-1.00e8 + 5.79e7i)T^{2} \)
43 \( 1 + (-1.29e4 + 7.45e3i)T + (7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + 2.29e8iT^{2} \)
53 \( 1 - 4.18e8T^{2} \)
59 \( 1 + (-6.19e8 - 3.57e8i)T^{2} \)
61 \( 1 + (-5.86e3 - 1.01e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.70e4 + 4.57e3i)T + (1.16e9 - 6.75e8i)T^{2} \)
71 \( 1 + (1.56e9 + 9.02e8i)T^{2} \)
73 \( 1 + (6.19e4 - 6.19e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 6.36e4T + 3.07e9T^{2} \)
83 \( 1 - 3.93e9iT^{2} \)
89 \( 1 + (4.83e9 - 2.79e9i)T^{2} \)
97 \( 1 + (-4.21e4 + 1.57e5i)T + (-7.43e9 - 4.29e9i)T^{2} \)
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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

−14.93624956361308702462330804942, −13.77763960202697507178279042505, −12.61157453802062035214063791896, −11.41474013990318464259889535611, −10.04828322123922611278057227241, −8.381608189521518017463181715154, −7.03316893730110295656910560284, −5.90314072602546503192319333002, −3.06204821997749464772072556703, −1.28793359170941709944000399240, 2.56260894961590219076026307063, 4.10993047835348550462897384659, 6.25264940520821159253750882879, 7.917996733911339671591758033125, 9.202996906358887933493286246313, 10.63176940926747029502874466823, 11.65126006686597080775690016665, 13.16349944999234674331611680311, 14.48621716545460432848212463048, 15.73925578919586881309298554351

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