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Properties

Label 2-39-39.8-c3-0-5
Degree $2$
Conductor $39$
Sign $0.215 + 0.976i$
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 + (11.6 − 20.8i)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 + (7.45 − 13.2i)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 + (0.795 − 1.41i)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 + (0.128 − 0.228i)15-s − 0.0503·16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.215 + 0.976i$
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: \((2,\ 39,\ (\ :3/2),\ 0.215 + 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0960436 - 0.0771315i\)
\(L(\frac12)\) \(\approx\) \(0.0960436 - 0.0771315i\)
\(L(\frac{5}{2})\) 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

−15.65927501720359445214666232488, −15.08035633063314785446297525542, −13.25019882113077509087677726126, −11.33172370864727107764183012002, −10.44709116386733099562797575243, −9.066092033286433734257486543317, −7.59238569558112365852403642225, −6.50920403966614286214672726225, −4.97641540431228701268123357133, −0.16037636950109806984896618463, 2.06636571142717413055663527514, 4.97629967953162032153790347070, 7.26643860677642544066925370627, 8.668727844650811113699844343115, 10.19988488230289251344391636709, 10.94906623637593761458465438669, 12.23916160106935515873842747174, 12.73984628381978714159200306149, 15.07680485927149397040872055400, 16.51330828679024922750315851661

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