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Properties

Label 2-105-105.53-c1-0-8
Degree 22
Conductor 105105
Sign 0.873+0.487i0.873 + 0.487i
Analytic cond. 0.8384290.838429
Root an. cond. 0.9156570.915657
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  + (0.0799 − 0.298i)2-s + (0.540 − 1.64i)3-s + (1.64 + 0.952i)4-s + (0.596 + 2.15i)5-s + (−0.447 − 0.292i)6-s + (−2.46 − 0.951i)7-s + (0.852 − 0.852i)8-s + (−2.41 − 1.77i)9-s + (0.690 − 0.00558i)10-s + (0.660 + 0.381i)11-s + (2.45 − 2.19i)12-s + (−2.27 − 2.27i)13-s + (−0.481 + 0.660i)14-s + (3.86 + 0.184i)15-s + (1.71 + 2.97i)16-s + (−4.69 + 1.25i)17-s + ⋯
L(s)  = 1  + (0.0565 − 0.210i)2-s + (0.312 − 0.950i)3-s + (0.824 + 0.476i)4-s + (0.266 + 0.963i)5-s + (−0.182 − 0.119i)6-s + (−0.933 − 0.359i)7-s + (0.301 − 0.301i)8-s + (−0.805 − 0.593i)9-s + (0.218 − 0.00176i)10-s + (0.199 + 0.114i)11-s + (0.709 − 0.634i)12-s + (−0.629 − 0.629i)13-s + (−0.128 + 0.176i)14-s + (0.998 + 0.0476i)15-s + (0.429 + 0.744i)16-s + (−1.13 + 0.305i)17-s + ⋯

Functional equation

Λ(s)=(105s/2 ΓC(s) L(s)=((0.873+0.487i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(105s/2 ΓC(s+1/2) L(s)=((0.873+0.487i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{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: 105105    =    3⋅5⋅73 \cdot 5 \cdot 7
Sign: 0.873+0.487i0.873 + 0.487i
Analytic conductor: 0.8384290.838429
Root analytic conductor: 0.9156570.915657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ105(53,⋅)\chi_{105} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 105, ( :1/2), 0.873+0.487i)(2,\ 105,\ (\ :1/2),\ 0.873 + 0.487i)

Particular Values

L(1)L(1) ≈\approx 1.20091−0.312429i1.20091 - 0.312429i
L(12)L(\frac12) ≈\approx 1.20091−0.312429i1.20091 - 0.312429i
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+(−0.540+1.64i)T 1 + (-0.540 + 1.64i)T
5 1+(−0.596−2.15i)T 1 + (-0.596 - 2.15i)T
7 1+(2.46+0.951i)T 1 + (2.46 + 0.951i)T
good2 1+(−0.0799+0.298i)T+(−1.73−i)T2 1 + (-0.0799 + 0.298i)T + (-1.73 - i)T^{2}
11 1+(−0.660−0.381i)T+(5.5+9.52i)T2 1 + (-0.660 - 0.381i)T + (5.5 + 9.52i)T^{2}
13 1+(2.27+2.27i)T+13iT2 1 + (2.27 + 2.27i)T + 13iT^{2}
17 1+(4.69−1.25i)T+(14.7−8.5i)T2 1 + (4.69 - 1.25i)T + (14.7 - 8.5i)T^{2}
19 1+(1.41−0.818i)T+(9.5−16.4i)T2 1 + (1.41 - 0.818i)T + (9.5 - 16.4i)T^{2}
23 1+(−7.39−1.98i)T+(19.9+11.5i)T2 1 + (-7.39 - 1.98i)T + (19.9 + 11.5i)T^{2}
29 1+4.94T+29T2 1 + 4.94T + 29T^{2}
31 1+(−2.96+5.13i)T+(−15.5−26.8i)T2 1 + (-2.96 + 5.13i)T + (-15.5 - 26.8i)T^{2}
37 1+(−3.41−0.915i)T+(32.0+18.5i)T2 1 + (-3.41 - 0.915i)T + (32.0 + 18.5i)T^{2}
41 1−4.35iT−41T2 1 - 4.35iT - 41T^{2}
43 1+(−2.69−2.69i)T+43iT2 1 + (-2.69 - 2.69i)T + 43iT^{2}
47 1+(−1.10+4.14i)T+(−40.7−23.5i)T2 1 + (-1.10 + 4.14i)T + (-40.7 - 23.5i)T^{2}
53 1+(1.79+6.71i)T+(−45.8+26.5i)T2 1 + (1.79 + 6.71i)T + (-45.8 + 26.5i)T^{2}
59 1+(−3.84+6.65i)T+(−29.5−51.0i)T2 1 + (-3.84 + 6.65i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.19+3.80i)T+(−30.5+52.8i)T2 1 + (2.19 + 3.80i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.0126−0.0471i)T+(−58.0+33.5i)T2 1 + (-0.0126 - 0.0471i)T + (-58.0 + 33.5i)T^{2}
71 1+12.4iT−71T2 1 + 12.4iT - 71T^{2}
73 1+(1.34−0.359i)T+(63.2−36.5i)T2 1 + (1.34 - 0.359i)T + (63.2 - 36.5i)T^{2}
79 1+(−3.66+2.11i)T+(39.5−68.4i)T2 1 + (-3.66 + 2.11i)T + (39.5 - 68.4i)T^{2}
83 1+(5.05−5.05i)T−83iT2 1 + (5.05 - 5.05i)T - 83iT^{2}
89 1+(0.453+0.785i)T+(−44.5+77.0i)T2 1 + (0.453 + 0.785i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.73+3.73i)T−97iT2 1 + (-3.73 + 3.73i)T - 97iT^{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

−13.25990574135713172067486611127, −12.91926529430470319018353159975, −11.60684736859995403691338299752, −10.74004565413189057121571730964, −9.456995753487573712577871588462, −7.80174373170695742902958436374, −6.92593539995165889238527396882, −6.23670208026834130883249928345, −3.42333967823018597516588761266, −2.36346477388012276580321747064, 2.56624740763306355575309366941, 4.54165344404402797406189813979, 5.70454355536339888806825482306, 6.98524833683424819563357942755, 8.823837211170176472227288131686, 9.415391442225670646725080961264, 10.57182402649436770339405091288, 11.65323785234021771127177527259, 12.87634642352758084308708352960, 14.02182893278758026657414443178

Graph of the ZZ-function along the critical line