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Properties

Label 2-45-45.2-c1-0-2
Degree 22
Conductor 4545
Sign 0.621+0.783i0.621 + 0.783i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
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  + (−0.430 − 1.60i)2-s + (1.35 + 1.08i)3-s + (−0.661 + 0.382i)4-s + (−2.23 + 0.154i)5-s + (1.15 − 2.63i)6-s + (−1.73 + 0.465i)7-s + (−1.45 − 1.45i)8-s + (0.661 + 2.92i)9-s + (1.20 + 3.51i)10-s + (3.12 + 1.80i)11-s + (−1.30 − 0.198i)12-s + (−1.27 − 0.342i)13-s + (1.49 + 2.59i)14-s + (−3.18 − 2.20i)15-s + (−2.47 + 4.28i)16-s + (0.277 − 0.277i)17-s + ⋯
L(s)  = 1  + (−0.304 − 1.13i)2-s + (0.781 + 0.624i)3-s + (−0.330 + 0.191i)4-s + (−0.997 + 0.0690i)5-s + (0.471 − 1.07i)6-s + (−0.656 + 0.175i)7-s + (−0.513 − 0.513i)8-s + (0.220 + 0.975i)9-s + (0.381 + 1.11i)10-s + (0.942 + 0.544i)11-s + (−0.377 − 0.0573i)12-s + (−0.354 − 0.0950i)13-s + (0.399 + 0.692i)14-s + (−0.822 − 0.568i)15-s + (−0.618 + 1.07i)16-s + (0.0671 − 0.0671i)17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.621+0.783i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1/2) L(s)=((0.621+0.783i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.621+0.783i0.621 + 0.783i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ45(2,⋅)\chi_{45} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1/2), 0.621+0.783i)(2,\ 45,\ (\ :1/2),\ 0.621 + 0.783i)

Particular Values

L(1)L(1) ≈\approx 0.692211−0.334449i0.692211 - 0.334449i
L(12)L(\frac12) ≈\approx 0.692211−0.334449i0.692211 - 0.334449i
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+(−1.35−1.08i)T 1 + (-1.35 - 1.08i)T
5 1+(2.23−0.154i)T 1 + (2.23 - 0.154i)T
good2 1+(0.430+1.60i)T+(−1.73+i)T2 1 + (0.430 + 1.60i)T + (-1.73 + i)T^{2}
7 1+(1.73−0.465i)T+(6.06−3.5i)T2 1 + (1.73 - 0.465i)T + (6.06 - 3.5i)T^{2}
11 1+(−3.12−1.80i)T+(5.5+9.52i)T2 1 + (-3.12 - 1.80i)T + (5.5 + 9.52i)T^{2}
13 1+(1.27+0.342i)T+(11.2+6.5i)T2 1 + (1.27 + 0.342i)T + (11.2 + 6.5i)T^{2}
17 1+(−0.277+0.277i)T−17iT2 1 + (-0.277 + 0.277i)T - 17iT^{2}
19 1+6.25iT−19T2 1 + 6.25iT - 19T^{2}
23 1+(−0.579+2.16i)T+(−19.9−11.5i)T2 1 + (-0.579 + 2.16i)T + (-19.9 - 11.5i)T^{2}
29 1+(1.56−2.71i)T+(−14.5−25.1i)T2 1 + (1.56 - 2.71i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.42+4.20i)T+(−15.5+26.8i)T2 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2}
37 1+(−5.55−5.55i)T+37iT2 1 + (-5.55 - 5.55i)T + 37iT^{2}
41 1+(−1.29+0.744i)T+(20.5−35.5i)T2 1 + (-1.29 + 0.744i)T + (20.5 - 35.5i)T^{2}
43 1+(1.10+4.10i)T+(−37.2+21.5i)T2 1 + (1.10 + 4.10i)T + (-37.2 + 21.5i)T^{2}
47 1+(1.02+3.82i)T+(−40.7+23.5i)T2 1 + (1.02 + 3.82i)T + (-40.7 + 23.5i)T^{2}
53 1+(−7.48−7.48i)T+53iT2 1 + (-7.48 - 7.48i)T + 53iT^{2}
59 1+(−0.279−0.483i)T+(−29.5+51.0i)T2 1 + (-0.279 - 0.483i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.96−5.13i)T+(−30.5−52.8i)T2 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.90−10.8i)T+(−58.0−33.5i)T2 1 + (2.90 - 10.8i)T + (-58.0 - 33.5i)T^{2}
71 1+8.01iT−71T2 1 + 8.01iT - 71T^{2}
73 1+(1.29−1.29i)T−73iT2 1 + (1.29 - 1.29i)T - 73iT^{2}
79 1+(−6.96−4.02i)T+(39.5+68.4i)T2 1 + (-6.96 - 4.02i)T + (39.5 + 68.4i)T^{2}
83 1+(0.560−0.150i)T+(71.8−41.5i)T2 1 + (0.560 - 0.150i)T + (71.8 - 41.5i)T^{2}
89 1+16.4T+89T2 1 + 16.4T + 89T^{2}
97 1+(−5.14+1.37i)T+(84.0−48.5i)T2 1 + (-5.14 + 1.37i)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

−15.48814334931586089528425035668, −14.84401014764592595659853185392, −13.11617539726927643038549088056, −11.97751315480468378099664596534, −10.92934913075462471574604889698, −9.699828025443930919117313866918, −8.839942659986665334564159960456, −7.05478013129401502391818272930, −4.21548998350390319370472016168, −2.86093392629714111854334407398, 3.55540741718480021542886524616, 6.25278543188853638197576661280, 7.35680509729583724471517328201, 8.267373598673132246039176657071, 9.398619374399456357120601865111, 11.61150115226788427953733667974, 12.65047593048595670642384387924, 14.21957815217478921101767747719, 14.92090090360207578431044602671, 16.09261190134310147601321615436

Graph of the ZZ-function along the critical line