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Properties

Label 2-39-39.32-c1-0-1
Degree $2$
Conductor $39$
Sign $0.960 + 0.277i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)3-s + (−1.73 + i)4-s + (−0.767 + 2.86i)7-s + (−1.5 − 2.59i)9-s + 3.46i·12-s + (−2.59 − 2.5i)13-s + (1.99 − 3.46i)16-s + (7.83 + 2.09i)19-s + (3.63 + 3.63i)21-s − 5i·25-s − 5.19·27-s + (−1.53 − 5.73i)28-s + (−7.83 + 7.83i)31-s + (5.19 + 3i)36-s + (2.09 − 0.562i)37-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (−0.866 + 0.5i)4-s + (−0.290 + 1.08i)7-s + (−0.5 − 0.866i)9-s + 0.999i·12-s + (−0.720 − 0.693i)13-s + (0.499 − 0.866i)16-s + (1.79 + 0.481i)19-s + (0.792 + 0.792i)21-s − i·25-s − 1.00·27-s + (−0.290 − 1.08i)28-s + (−1.40 + 1.40i)31-s + (0.866 + 0.5i)36-s + (0.344 − 0.0924i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734633 - 0.103967i\)
\(L(\frac12)\) \(\approx\) \(0.734633 - 0.103967i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 2.5i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + (0.767 - 2.86i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.83 - 2.09i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.83 - 7.83i)T - 31iT^{2} \)
37 \( 1 + (-2.09 + 0.562i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.33 - 7.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.205 - 0.767i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.36 + 9.36i)T + 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (16.4 + 4.40i)T + (84.0 + 48.5i)T^{2} \)
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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

−16.28138119126810073270275984764, −14.80180401531769427686703023050, −13.81765315447578573887117314647, −12.60526945583769593725335756279, −12.04244590137973943733356661932, −9.662357628454430853550576199998, −8.614704473648495175511787933697, −7.45618645320222714468884015368, −5.50669007436295339404409935203, −3.07843900728166971573586321345, 3.82871138839055860330868972501, 5.18384665668504267031376994747, 7.49888675486433736155097520690, 9.266331349291400128893358857922, 9.878455700632552780346137377328, 11.20671029059133093468554275184, 13.28427303704086341610784007970, 14.04727720450050410999165257189, 14.99871328999823903849859673460, 16.35775847699612248024532957123

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