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Properties

Label 2-39-39.32-c1-0-1
Degree 22
Conductor 3939
Sign 0.960+0.277i0.960 + 0.277i
Analytic cond. 0.3114160.311416
Root an. cond. 0.5580470.558047
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(39s/2 ΓC(s) L(s)=((0.960+0.277i) Λ‾(2−s)\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}
Λ(s)=(39s/2 ΓC(s+1/2) L(s)=((0.960+0.277i) Λ‾(1−s)\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: 22
Conductor: 3939    =    3⋅133 \cdot 13
Sign: 0.960+0.277i0.960 + 0.277i
Analytic conductor: 0.3114160.311416
Root analytic conductor: 0.5580470.558047
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ39(32,⋅)\chi_{39} (32, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 39, ( :1/2), 0.960+0.277i)(2,\ 39,\ (\ :1/2),\ 0.960 + 0.277i)

Particular Values

L(1)L(1) ≈\approx 0.734633−0.103967i0.734633 - 0.103967i
L(12)L(\frac12) ≈\approx 0.734633−0.103967i0.734633 - 0.103967i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.866+1.5i)T 1 + (-0.866 + 1.5i)T
13 1+(2.59+2.5i)T 1 + (2.59 + 2.5i)T
good2 1+(1.73−i)T2 1 + (1.73 - i)T^{2}
5 1+5iT2 1 + 5iT^{2}
7 1+(0.767−2.86i)T+(−6.06−3.5i)T2 1 + (0.767 - 2.86i)T + (-6.06 - 3.5i)T^{2}
11 1+(−9.52+5.5i)T2 1 + (-9.52 + 5.5i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(−7.83−2.09i)T+(16.4+9.5i)T2 1 + (-7.83 - 2.09i)T + (16.4 + 9.5i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(14.5+25.1i)T2 1 + (14.5 + 25.1i)T^{2}
31 1+(7.83−7.83i)T−31iT2 1 + (7.83 - 7.83i)T - 31iT^{2}
37 1+(−2.09+0.562i)T+(32.0−18.5i)T2 1 + (-2.09 + 0.562i)T + (32.0 - 18.5i)T^{2}
41 1+(35.5−20.5i)T2 1 + (35.5 - 20.5i)T^{2}
43 1+(1.5−0.866i)T+(21.5−37.2i)T2 1 + (1.5 - 0.866i)T + (21.5 - 37.2i)T^{2}
47 1−47iT2 1 - 47iT^{2}
53 1−53T2 1 - 53T^{2}
59 1+(51.0+29.5i)T2 1 + (51.0 + 29.5i)T^{2}
61 1+(−4.33−7.5i)T+(−30.5+52.8i)T2 1 + (-4.33 - 7.5i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.205−0.767i)T+(−58.0+33.5i)T2 1 + (-0.205 - 0.767i)T + (-58.0 + 33.5i)T^{2}
71 1+(−61.4−35.5i)T2 1 + (-61.4 - 35.5i)T^{2}
73 1+(9.36+9.36i)T+73iT2 1 + (9.36 + 9.36i)T + 73iT^{2}
79 1−12.1T+79T2 1 - 12.1T + 79T^{2}
83 1+83iT2 1 + 83iT^{2}
89 1+(−77.0+44.5i)T2 1 + (-77.0 + 44.5i)T^{2}
97 1+(16.4+4.40i)T+(84.0+48.5i)T2 1 + (16.4 + 4.40i)T + (84.0 + 48.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line