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Properties

Label 2-39-13.12-c3-0-6
Degree $2$
Conductor $39$
Sign $-0.719 + 0.694i$
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  − 4.54i·2-s + 3·3-s − 12.6·4-s − 12.9i·5-s − 13.6i·6-s + 16.7i·7-s + 21.2i·8-s + 9·9-s − 58.7·10-s + 24.9i·11-s − 38.0·12-s + (33.7 − 32.5i)13-s + 76.0·14-s − 38.7i·15-s − 4.67·16-s + 134.·17-s + ⋯
L(s)  = 1  − 1.60i·2-s + 0.577·3-s − 1.58·4-s − 1.15i·5-s − 0.928i·6-s + 0.903i·7-s + 0.940i·8-s + 0.333·9-s − 1.85·10-s + 0.683i·11-s − 0.915·12-s + (0.719 − 0.694i)13-s + 1.45·14-s − 0.666i·15-s − 0.0731·16-s + 1.91·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.538510 - 1.33274i\)
\(L(\frac12)\) \(\approx\) \(0.538510 - 1.33274i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 + (-33.7 + 32.5i)T \)
good2 \( 1 + 4.54iT - 8T^{2} \)
5 \( 1 + 12.9iT - 125T^{2} \)
7 \( 1 - 16.7iT - 343T^{2} \)
11 \( 1 - 24.9iT - 1.33e3T^{2} \)
17 \( 1 - 134.T + 4.91e3T^{2} \)
19 \( 1 - 14.9iT - 6.85e3T^{2} \)
23 \( 1 + 72T + 1.21e4T^{2} \)
29 \( 1 + 206.T + 2.43e4T^{2} \)
31 \( 1 - 249. iT - 2.97e4T^{2} \)
37 \( 1 - 293. iT - 5.06e4T^{2} \)
41 \( 1 + 250. iT - 6.89e4T^{2} \)
43 \( 1 + 432.T + 7.95e4T^{2} \)
47 \( 1 + 159. iT - 1.03e5T^{2} \)
53 \( 1 + 194.T + 1.48e5T^{2} \)
59 \( 1 - 232. iT - 2.05e5T^{2} \)
61 \( 1 + 185.T + 2.26e5T^{2} \)
67 \( 1 - 39.4iT - 3.00e5T^{2} \)
71 \( 1 + 920. iT - 3.57e5T^{2} \)
73 \( 1 + 549. iT - 3.89e5T^{2} \)
79 \( 1 - 933.T + 4.93e5T^{2} \)
83 \( 1 - 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 + 532. iT - 7.04e5T^{2} \)
97 \( 1 - 362. iT - 9.12e5T^{2} \)
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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

−15.14481611646285590727839823813, −13.61868038440777728690315291038, −12.49543135788814011096897349874, −12.07250784738461415991620196438, −10.29394733648041933706688208559, −9.271735078580329713532904555382, −8.223636390196996803376716728956, −5.19839155320078868822551234660, −3.41298872181280410099677272703, −1.52752033709582105481295709391, 3.73023945591655165223810402365, 5.96935457729125878076055408982, 7.18872780027925338184984574409, 8.058888026448256052957670677929, 9.670130327916787056147022595314, 11.14759597093676363984320262255, 13.44779215954998479655034817859, 14.25184112954517944522757115429, 14.84182106751255101651434510221, 16.18951110444129537535638517948

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