↑

Properties

Label 2-650-65.32-c1-0-3
Degree 22
Conductor 650650
Sign −0.500+0.865i-0.500 + 0.865i
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.876 + 3.27i)3-s + (−0.499 − 0.866i)4-s + (−3.27 − 0.876i)6-s + (−3.61 + 2.08i)7-s + 0.999·8-s + (−7.33 + 4.23i)9-s + (−0.732 + 0.196i)11-s + (2.39 − 2.39i)12-s + (3.59 + 0.333i)13-s − 4.17i·14-s + (−0.5 + 0.866i)16-s + (2.47 + 0.662i)17-s − 8.46i·18-s + (1.24 − 4.66i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.505 + 1.88i)3-s + (−0.249 − 0.433i)4-s + (−1.33 − 0.357i)6-s + (−1.36 + 0.789i)7-s + 0.353·8-s + (−2.44 + 1.41i)9-s + (−0.220 + 0.0591i)11-s + (0.691 − 0.691i)12-s + (0.995 + 0.0924i)13-s − 1.11i·14-s + (−0.125 + 0.216i)16-s + (0.599 + 0.160i)17-s − 1.99i·18-s + (0.286 − 1.07i)19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.500+0.865i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((−0.500+0.865i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.500+0.865i-0.500 + 0.865i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(357,⋅)\chi_{650} (357, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), −0.500+0.865i)(2,\ 650,\ (\ :1/2),\ -0.500 + 0.865i)

Particular Values

L(1)L(1) ≈\approx 0.423379−0.733746i0.423379 - 0.733746i
L(12)L(\frac12) ≈\approx 0.423379−0.733746i0.423379 - 0.733746i
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
5 1 1
13 1+(−3.59−0.333i)T 1 + (-3.59 - 0.333i)T
good3 1+(−0.876−3.27i)T+(−2.59+1.5i)T2 1 + (-0.876 - 3.27i)T + (-2.59 + 1.5i)T^{2}
7 1+(3.61−2.08i)T+(3.5−6.06i)T2 1 + (3.61 - 2.08i)T + (3.5 - 6.06i)T^{2}
11 1+(0.732−0.196i)T+(9.52−5.5i)T2 1 + (0.732 - 0.196i)T + (9.52 - 5.5i)T^{2}
17 1+(−2.47−0.662i)T+(14.7+8.5i)T2 1 + (-2.47 - 0.662i)T + (14.7 + 8.5i)T^{2}
19 1+(−1.24+4.66i)T+(−16.4−9.5i)T2 1 + (-1.24 + 4.66i)T + (-16.4 - 9.5i)T^{2}
23 1+(2.01−0.538i)T+(19.9−11.5i)T2 1 + (2.01 - 0.538i)T + (19.9 - 11.5i)T^{2}
29 1+(−4.35−2.51i)T+(14.5+25.1i)T2 1 + (-4.35 - 2.51i)T + (14.5 + 25.1i)T^{2}
31 1+(1.70−1.70i)T−31iT2 1 + (1.70 - 1.70i)T - 31iT^{2}
37 1+(1.55+0.895i)T+(18.5+32.0i)T2 1 + (1.55 + 0.895i)T + (18.5 + 32.0i)T^{2}
41 1+(−0.417−1.55i)T+(−35.5+20.5i)T2 1 + (-0.417 - 1.55i)T + (-35.5 + 20.5i)T^{2}
43 1+(2.44−9.11i)T+(−37.2−21.5i)T2 1 + (2.44 - 9.11i)T + (-37.2 - 21.5i)T^{2}
47 1+4.89iT−47T2 1 + 4.89iT - 47T^{2}
53 1+(3.87−3.87i)T−53iT2 1 + (3.87 - 3.87i)T - 53iT^{2}
59 1+(12.6+3.38i)T+(51.0+29.5i)T2 1 + (12.6 + 3.38i)T + (51.0 + 29.5i)T^{2}
61 1+(0.00936+0.0162i)T+(−30.5+52.8i)T2 1 + (0.00936 + 0.0162i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.67−9.83i)T+(−33.5−58.0i)T2 1 + (5.67 - 9.83i)T + (-33.5 - 58.0i)T^{2}
71 1+(−7.24−1.94i)T+(61.4+35.5i)T2 1 + (-7.24 - 1.94i)T + (61.4 + 35.5i)T^{2}
73 1−2.26T+73T2 1 - 2.26T + 73T^{2}
79 1−4.98iT−79T2 1 - 4.98iT - 79T^{2}
83 1−2.56iT−83T2 1 - 2.56iT - 83T^{2}
89 1+(1.73+6.47i)T+(−77.0+44.5i)T2 1 + (1.73 + 6.47i)T + (-77.0 + 44.5i)T^{2}
97 1+(−3.50−6.07i)T+(−48.5+84.0i)T2 1 + (-3.50 - 6.07i)T + (-48.5 + 84.0i)T^{2}
show more
show less
   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

−10.69097435077110838290662756554, −10.02093963229037719608769934365, −9.292210356292972338708467329996, −8.873437214867278851135802128828, −7.993212426000721102777529272059, −6.46705435411443742045520714936, −5.66351033767313242356182707804, −4.77588564374347727819847479608, −3.58045972108460270638700545108, −2.82974725373952884791402001797, 0.48121120679471329072528673216, 1.65606775874990892625134832744, 3.02373242375398201316576486104, 3.65234577591582624964707932443, 5.89856376146258185304174270051, 6.54483898937082737505716682486, 7.49893983189155888223252656192, 8.087229637145619624742440740260, 9.011254652404727389088339624910, 9.937845454682083706874772519474

Graph of the ZZ-function along the critical line