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Properties

Label 2-39-39.8-c3-0-11
Degree $2$
Conductor $39$
Sign $-0.324 + 0.945i$
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  + (3.24 − 3.24i)2-s + (−5.00 − 1.40i)3-s − 13.1i·4-s + (2.07 − 2.07i)5-s + (−20.8 + 11.6i)6-s + (7.93 − 7.93i)7-s + (−16.6 − 16.6i)8-s + (23.0 + 14.0i)9-s − 13.4i·10-s + (33.2 + 33.2i)11-s + (−18.4 + 65.6i)12-s + (−45.5 + 11.0i)13-s − 51.5i·14-s + (−13.2 + 7.45i)15-s − 3.22·16-s + 80.7·17-s + ⋯
L(s)  = 1  + (1.14 − 1.14i)2-s + (−0.962 − 0.270i)3-s − 1.64i·4-s + (0.185 − 0.185i)5-s + (−1.41 + 0.795i)6-s + (0.428 − 0.428i)7-s + (−0.735 − 0.735i)8-s + (0.853 + 0.520i)9-s − 0.426i·10-s + (0.911 + 0.911i)11-s + (−0.443 + 1.57i)12-s + (−0.971 + 0.235i)13-s − 0.984i·14-s + (−0.228 + 0.128i)15-s − 0.0503·16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.324 + 0.945i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.324 + 0.945i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.04666 - 1.46546i\)
\(L(\frac12)\) \(\approx\) \(1.04666 - 1.46546i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.00 + 1.40i)T \)
13 \( 1 + (45.5 - 11.0i)T \)
good2 \( 1 + (-3.24 + 3.24i)T - 8iT^{2} \)
5 \( 1 + (-2.07 + 2.07i)T - 125iT^{2} \)
7 \( 1 + (-7.93 + 7.93i)T - 343iT^{2} \)
11 \( 1 + (-33.2 - 33.2i)T + 1.33e3iT^{2} \)
17 \( 1 - 80.7T + 4.91e3T^{2} \)
19 \( 1 + (66.4 + 66.4i)T + 6.85e3iT^{2} \)
23 \( 1 + 131.T + 1.21e4T^{2} \)
29 \( 1 - 240. iT - 2.43e4T^{2} \)
31 \( 1 + (68.7 + 68.7i)T + 2.97e4iT^{2} \)
37 \( 1 + (162. - 162. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-217. + 217. i)T - 6.89e4iT^{2} \)
43 \( 1 + 235. iT - 7.95e4T^{2} \)
47 \( 1 + (38.1 + 38.1i)T + 1.03e5iT^{2} \)
53 \( 1 + 77.2iT - 1.48e5T^{2} \)
59 \( 1 + (-253. - 253. i)T + 2.05e5iT^{2} \)
61 \( 1 - 93.4T + 2.26e5T^{2} \)
67 \( 1 + (226. + 226. i)T + 3.00e5iT^{2} \)
71 \( 1 + (-122. + 122. i)T - 3.57e5iT^{2} \)
73 \( 1 + (1.90 - 1.90i)T - 3.89e5iT^{2} \)
79 \( 1 - 145.T + 4.93e5T^{2} \)
83 \( 1 + (592. - 592. i)T - 5.71e5iT^{2} \)
89 \( 1 + (386. + 386. i)T + 7.04e5iT^{2} \)
97 \( 1 + (747. + 747. i)T + 9.12e5iT^{2} \)
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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

−14.90977180558863601338944492133, −13.92660576048479549499915222860, −12.58547311995245416898979364669, −12.05413328607792302194551733773, −10.90919796193194981253801818012, −9.819591589757603787969561256635, −7.14548129833448895872593318515, −5.37294034438577093794123047179, −4.24937276381927460840969964818, −1.67246852332614545632307548662, 4.07960527876186969950625710132, 5.56443548256209788332559873341, 6.35395151946553095991296450016, 7.955768633560120394735961295037, 10.04928742913787021299917232414, 11.75768936296948794525991525790, 12.55038402976779475047925632905, 14.16837224003291099536286221824, 14.81052118865343553528078041662, 16.10795060971430580555605091927

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