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Properties

Label 2-39-13.4-c3-0-5
Degree $2$
Conductor $39$
Sign $-0.0876 + 0.996i$
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  + (−0.794 − 0.458i)2-s + (−1.5 + 2.59i)3-s + (−3.57 − 6.19i)4-s − 15.4i·5-s + (2.38 − 1.37i)6-s + (17.8 − 10.2i)7-s + 13.9i·8-s + (−4.5 − 7.79i)9-s + (−7.09 + 12.2i)10-s + (−57.0 − 32.9i)11-s + 21.4·12-s + (19.2 + 42.7i)13-s − 18.8·14-s + (40.1 + 23.2i)15-s + (−22.2 + 38.5i)16-s + (22.1 + 38.3i)17-s + ⋯
L(s)  = 1  + (−0.280 − 0.162i)2-s + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)4-s − 1.38i·5-s + (0.162 − 0.0936i)6-s + (0.962 − 0.555i)7-s + 0.614i·8-s + (−0.166 − 0.288i)9-s + (−0.224 + 0.388i)10-s + (−1.56 − 0.902i)11-s + 0.516·12-s + (0.410 + 0.911i)13-s − 0.360·14-s + (0.691 + 0.399i)15-s + (−0.347 + 0.602i)16-s + (0.315 + 0.547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0876 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0876 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.0876 + 0.996i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.0876 + 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.601311 - 0.656520i\)
\(L(\frac12)\) \(\approx\) \(0.601311 - 0.656520i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-19.2 - 42.7i)T \)
good2 \( 1 + (0.794 + 0.458i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 15.4iT - 125T^{2} \)
7 \( 1 + (-17.8 + 10.2i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (57.0 + 32.9i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-22.1 - 38.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-127. + 73.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-26.5 + 46.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-19.3 + 33.4i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 88.3iT - 2.97e4T^{2} \)
37 \( 1 + (-68.3 - 39.4i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-307. - 177. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (203. + 353. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 67.9iT - 1.03e5T^{2} \)
53 \( 1 - 226.T + 1.48e5T^{2} \)
59 \( 1 + (123. - 71.0i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (133. + 231. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-356. - 205. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (79.2 - 45.7i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 63.1iT - 3.89e5T^{2} \)
79 \( 1 + 287.T + 4.93e5T^{2} \)
83 \( 1 + 373. iT - 5.71e5T^{2} \)
89 \( 1 + (-103. - 59.7i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-480. + 277. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.67857196282702684238851085748, −14.12967935731342205392442124704, −13.24201482790509997984960101692, −11.48625806762695343999042951012, −10.52023560750634584378331422132, −9.157941997819156663537096942368, −8.155671278959517453120984771507, −5.46515044999546235078040469955, −4.65949559279352466068950645809, −0.935924330830443060854215264927, 2.87978335562636240717531867973, 5.37364255181741869840176736858, 7.41418082332163308747254874937, 7.935786800214942753600980348807, 9.962773696412015119886680144182, 11.25776597217845788226500007946, 12.45140546765481924569025698391, 13.66782585520119773515485749294, 14.92101441997457446512842786497, 16.02944570414291138275490196390

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