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Properties

Label 2-39-13.6-c2-0-0
Degree $2$
Conductor $39$
Sign $-0.0789 - 0.996i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.613 + 2.29i)2-s + (−0.866 + 1.5i)3-s + (−1.40 + 0.810i)4-s + (−1.21 + 1.21i)5-s + (−3.96 − 1.06i)6-s + (1.50 − 5.62i)7-s + (3.98 + 3.98i)8-s + (−1.5 − 2.59i)9-s + (−3.52 − 2.03i)10-s + (4.43 − 1.18i)11-s − 2.80i·12-s + (5.13 − 11.9i)13-s + 13.8·14-s + (−0.769 − 2.87i)15-s + (−9.92 + 17.1i)16-s + (3.18 − 1.84i)17-s + ⋯
L(s)  = 1  + (0.306 + 1.14i)2-s + (−0.288 + 0.5i)3-s + (−0.350 + 0.202i)4-s + (−0.242 + 0.242i)5-s + (−0.661 − 0.177i)6-s + (0.215 − 0.803i)7-s + (0.498 + 0.498i)8-s + (−0.166 − 0.288i)9-s + (−0.352 − 0.203i)10-s + (0.402 − 0.107i)11-s − 0.233i·12-s + (0.394 − 0.918i)13-s + 0.986·14-s + (−0.0513 − 0.191i)15-s + (−0.620 + 1.07i)16-s + (0.187 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.0789 - 0.996i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ -0.0789 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.795022 + 0.860495i\)
\(L(\frac12)\) \(\approx\) \(0.795022 + 0.860495i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 1.5i)T \)
13 \( 1 + (-5.13 + 11.9i)T \)
good2 \( 1 + (-0.613 - 2.29i)T + (-3.46 + 2i)T^{2} \)
5 \( 1 + (1.21 - 1.21i)T - 25iT^{2} \)
7 \( 1 + (-1.50 + 5.62i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-4.43 + 1.18i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-3.18 + 1.84i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (20.4 + 5.47i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (32.1 + 18.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (24.5 - 42.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-16.7 + 16.7i)T - 961iT^{2} \)
37 \( 1 + (-48.3 + 12.9i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (2.52 + 9.41i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (7.44 - 4.29i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (50.5 + 50.5i)T + 2.20e3iT^{2} \)
53 \( 1 - 32.0T + 2.80e3T^{2} \)
59 \( 1 + (3.88 - 14.5i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-50.6 - 87.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-17.6 - 65.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-91.2 - 24.4i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (61.3 + 61.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 128.T + 6.24e3T^{2} \)
83 \( 1 + (0.0998 - 0.0998i)T - 6.88e3iT^{2} \)
89 \( 1 + (109. - 29.3i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-45.7 - 12.2i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−16.30720718573257547274148068988, −15.14210376190837810589131778216, −14.44293437767145342695418680167, −13.13175375478338936181310446761, −11.28991904666711088430391224152, −10.34076530455769903680896496531, −8.376855047642531651510779069755, −7.07906771723573799069626000657, −5.75404185204452917209957699487, −4.15706526152139468190056892150, 2.00100864246532679916516909613, 4.20404966055548007345096185389, 6.27988546117402270706333354727, 8.132492699664229733999674415682, 9.776845676926636774746847851079, 11.37430866403046080657597220975, 11.94310090937654421211307624334, 12.93394919442004885931817415456, 14.17142314652806038919238571496, 15.75633258543913685412220388359

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