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Properties

Label 2-54-27.7-c1-0-1
Degree 22
Conductor 5454
Sign 0.873+0.487i0.873 + 0.487i
Analytic cond. 0.4311920.431192
Root an. cond. 0.6566520.656652
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  + (−0.939 + 0.342i)2-s + (0.552 − 1.64i)3-s + (0.766 − 0.642i)4-s + (−0.177 − 1.00i)5-s + (0.0419 + 1.73i)6-s + (2.04 + 1.71i)7-s + (−0.500 + 0.866i)8-s + (−2.38 − 1.81i)9-s + (0.510 + 0.884i)10-s + (−0.720 + 4.08i)11-s + (−0.631 − 1.61i)12-s + (−3.68 − 1.34i)13-s + (−2.50 − 0.912i)14-s + (−1.74 − 0.264i)15-s + (0.173 − 0.984i)16-s + (0.925 + 1.60i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.319 − 0.947i)3-s + (0.383 − 0.321i)4-s + (−0.0793 − 0.449i)5-s + (0.0171 + 0.706i)6-s + (0.772 + 0.647i)7-s + (−0.176 + 0.306i)8-s + (−0.796 − 0.604i)9-s + (0.161 + 0.279i)10-s + (−0.217 + 1.23i)11-s + (−0.182 − 0.465i)12-s + (−1.02 − 0.371i)13-s + (−0.669 − 0.243i)14-s + (−0.451 − 0.0684i)15-s + (0.0434 − 0.246i)16-s + (0.224 + 0.388i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.873+0.487i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1/2) L(s)=((0.873+0.487i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.873+0.487i0.873 + 0.487i
Analytic conductor: 0.4311920.431192
Root analytic conductor: 0.6566520.656652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ54(7,⋅)\chi_{54} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1/2), 0.873+0.487i)(2,\ 54,\ (\ :1/2),\ 0.873 + 0.487i)

Particular Values

L(1)L(1) ≈\approx 0.684092−0.177982i0.684092 - 0.177982i
L(12)L(\frac12) ≈\approx 0.684092−0.177982i0.684092 - 0.177982i
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)
bad2 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
3 1+(−0.552+1.64i)T 1 + (-0.552 + 1.64i)T
good5 1+(0.177+1.00i)T+(−4.69+1.71i)T2 1 + (0.177 + 1.00i)T + (-4.69 + 1.71i)T^{2}
7 1+(−2.04−1.71i)T+(1.21+6.89i)T2 1 + (-2.04 - 1.71i)T + (1.21 + 6.89i)T^{2}
11 1+(0.720−4.08i)T+(−10.3−3.76i)T2 1 + (0.720 - 4.08i)T + (-10.3 - 3.76i)T^{2}
13 1+(3.68+1.34i)T+(9.95+8.35i)T2 1 + (3.68 + 1.34i)T + (9.95 + 8.35i)T^{2}
17 1+(−0.925−1.60i)T+(−8.5+14.7i)T2 1 + (-0.925 - 1.60i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.21−5.57i)T+(−9.5−16.4i)T2 1 + (3.21 - 5.57i)T + (-9.5 - 16.4i)T^{2}
23 1+(−6.69+5.61i)T+(3.99−22.6i)T2 1 + (-6.69 + 5.61i)T + (3.99 - 22.6i)T^{2}
29 1+(1.17−0.428i)T+(22.2−18.6i)T2 1 + (1.17 - 0.428i)T + (22.2 - 18.6i)T^{2}
31 1+(2.56−2.15i)T+(5.38−30.5i)T2 1 + (2.56 - 2.15i)T + (5.38 - 30.5i)T^{2}
37 1+(4.58+7.94i)T+(−18.5+32.0i)T2 1 + (4.58 + 7.94i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.53+1.28i)T+(31.4+26.3i)T2 1 + (3.53 + 1.28i)T + (31.4 + 26.3i)T^{2}
43 1+(−0.536+3.04i)T+(−40.4−14.7i)T2 1 + (-0.536 + 3.04i)T + (-40.4 - 14.7i)T^{2}
47 1+(2.11+1.77i)T+(8.16+46.2i)T2 1 + (2.11 + 1.77i)T + (8.16 + 46.2i)T^{2}
53 1+0.231T+53T2 1 + 0.231T + 53T^{2}
59 1+(−0.613−3.48i)T+(−55.4+20.1i)T2 1 + (-0.613 - 3.48i)T + (-55.4 + 20.1i)T^{2}
61 1+(−0.405−0.339i)T+(10.5+60.0i)T2 1 + (-0.405 - 0.339i)T + (10.5 + 60.0i)T^{2}
67 1+(−7.67−2.79i)T+(51.3+43.0i)T2 1 + (-7.67 - 2.79i)T + (51.3 + 43.0i)T^{2}
71 1+(4.03+6.98i)T+(−35.5+61.4i)T2 1 + (4.03 + 6.98i)T + (-35.5 + 61.4i)T^{2}
73 1+(−1.57+2.72i)T+(−36.5−63.2i)T2 1 + (-1.57 + 2.72i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.43+0.886i)T+(60.5−50.7i)T2 1 + (-2.43 + 0.886i)T + (60.5 - 50.7i)T^{2}
83 1+(−7.55+2.74i)T+(63.5−53.3i)T2 1 + (-7.55 + 2.74i)T + (63.5 - 53.3i)T^{2}
89 1+(6.12−10.6i)T+(−44.5−77.0i)T2 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2}
97 1+(1.51−8.57i)T+(−91.1−33.1i)T2 1 + (1.51 - 8.57i)T + (-91.1 - 33.1i)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

−14.93429173397734085013910976492, −14.63899056561241445357673567825, −12.63432735249090237157230699214, −12.23056438020642826027399933696, −10.52907970405077909816942499740, −8.983127075841165661409172852424, −8.067362704092046075914041917458, −6.96114451524025263038274085798, −5.21576089903790852035786638133, −2.07713370685981269596023571491, 3.07807768430759731796366311785, 4.92801450494612008646801785527, 7.17638713467436565895528380467, 8.489307361361171263836544814333, 9.611936908246278466799754183946, 10.94257858274066432470952569801, 11.32033399922844689736679790448, 13.43476044182148212895669617862, 14.55736832960162331842961831637, 15.47463132599823605608320873180

Graph of the ZZ-function along the critical line