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Properties

Label 2-968-88.5-c1-0-84
Degree 22
Conductor 968968
Sign 0.999−0.0373i0.999 - 0.0373i
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
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  + (0.301 + 1.38i)2-s + (1.50 − 2.06i)3-s + (−1.81 + 0.832i)4-s + (2.56 + 0.832i)5-s + (3.30 + 1.45i)6-s + (3.19 − 2.32i)7-s + (−1.69 − 2.26i)8-s + (−1.08 − 3.35i)9-s + (−0.379 + 3.79i)10-s + (−1.01 + 5.00i)12-s + (−0.416 + 0.135i)13-s + (4.16 + 3.71i)14-s + (5.57 − 4.04i)15-s + (2.61 − 3.02i)16-s + (0.739 − 2.27i)17-s + (4.30 − 2.51i)18-s + ⋯
L(s)  = 1  + (0.212 + 0.977i)2-s + (0.867 − 1.19i)3-s + (−0.909 + 0.416i)4-s + (1.14 + 0.372i)5-s + (1.35 + 0.593i)6-s + (1.20 − 0.877i)7-s + (−0.600 − 0.799i)8-s + (−0.363 − 1.11i)9-s + (−0.119 + 1.19i)10-s + (−0.291 + 1.44i)12-s + (−0.115 + 0.0375i)13-s + (1.11 + 0.993i)14-s + (1.43 − 1.04i)15-s + (0.653 − 0.756i)16-s + (0.179 − 0.552i)17-s + (1.01 − 0.593i)18-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((0.999−0.0373i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0373i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((0.999−0.0373i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0373i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: 0.999−0.0373i0.999 - 0.0373i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(269,⋅)\chi_{968} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), 0.999−0.0373i)(2,\ 968,\ (\ :1/2),\ 0.999 - 0.0373i)

Particular Values

L(1)L(1) ≈\approx 2.83158+0.0528372i2.83158 + 0.0528372i
L(12)L(\frac12) ≈\approx 2.83158+0.0528372i2.83158 + 0.0528372i
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+(−0.301−1.38i)T 1 + (-0.301 - 1.38i)T
11 1 1
good3 1+(−1.50+2.06i)T+(−0.927−2.85i)T2 1 + (-1.50 + 2.06i)T + (-0.927 - 2.85i)T^{2}
5 1+(−2.56−0.832i)T+(4.04+2.93i)T2 1 + (-2.56 - 0.832i)T + (4.04 + 2.93i)T^{2}
7 1+(−3.19+2.32i)T+(2.16−6.65i)T2 1 + (-3.19 + 2.32i)T + (2.16 - 6.65i)T^{2}
13 1+(0.416−0.135i)T+(10.5−7.64i)T2 1 + (0.416 - 0.135i)T + (10.5 - 7.64i)T^{2}
17 1+(−0.739+2.27i)T+(−13.7−9.99i)T2 1 + (-0.739 + 2.27i)T + (-13.7 - 9.99i)T^{2}
19 1+(3.14−4.32i)T+(−5.87−18.0i)T2 1 + (3.14 - 4.32i)T + (-5.87 - 18.0i)T^{2}
23 1+1.35T+23T2 1 + 1.35T + 23T^{2}
29 1+(−0.0866−0.119i)T+(−8.96+27.5i)T2 1 + (-0.0866 - 0.119i)T + (-8.96 + 27.5i)T^{2}
31 1+(−0.489−1.50i)T+(−25.0+18.2i)T2 1 + (-0.489 - 1.50i)T + (-25.0 + 18.2i)T^{2}
37 1+(4.33+5.97i)T+(−11.4+35.1i)T2 1 + (4.33 + 5.97i)T + (-11.4 + 35.1i)T^{2}
41 1+(2.92+2.12i)T+(12.6+38.9i)T2 1 + (2.92 + 2.12i)T + (12.6 + 38.9i)T^{2}
43 1−6.62iT−43T2 1 - 6.62iT - 43T^{2}
47 1+(2.44+1.77i)T+(14.5+44.6i)T2 1 + (2.44 + 1.77i)T + (14.5 + 44.6i)T^{2}
53 1+(4.41−1.43i)T+(42.8−31.1i)T2 1 + (4.41 - 1.43i)T + (42.8 - 31.1i)T^{2}
59 1+(−7.35−10.1i)T+(−18.2+56.1i)T2 1 + (-7.35 - 10.1i)T + (-18.2 + 56.1i)T^{2}
61 1+(−0.627−0.203i)T+(49.3+35.8i)T2 1 + (-0.627 - 0.203i)T + (49.3 + 35.8i)T^{2}
67 1−3.00iT−67T2 1 - 3.00iT - 67T^{2}
71 1+(3.26−10.0i)T+(−57.4−41.7i)T2 1 + (3.26 - 10.0i)T + (-57.4 - 41.7i)T^{2}
73 1+(−8.61+6.25i)T+(22.5−69.4i)T2 1 + (-8.61 + 6.25i)T + (22.5 - 69.4i)T^{2}
79 1+(−1.99−6.13i)T+(−63.9+46.4i)T2 1 + (-1.99 - 6.13i)T + (-63.9 + 46.4i)T^{2}
83 1+(2.17+0.705i)T+(67.1+48.7i)T2 1 + (2.17 + 0.705i)T + (67.1 + 48.7i)T^{2}
89 1−11.5T+89T2 1 - 11.5T + 89T^{2}
97 1+(−2.68−8.25i)T+(−78.4+57.0i)T2 1 + (-2.68 - 8.25i)T + (-78.4 + 57.0i)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

−9.835813786920975478722577558203, −8.805469408912798664704181928111, −8.105604427372673008460351592572, −7.47357594477401157595230180057, −6.80719339899136729432813943992, −5.96416676849488658275637841703, −4.95819643291526828801356298986, −3.76817065838731964426169287029, −2.37604113480691310551152175981, −1.33626667343194804875101508290, 1.77231736825601048729061575442, 2.45026527461609864847138821998, 3.61164056641655881460520104704, 4.80760385299111195004796608425, 5.08154139300808654959361103798, 6.18319005284047350046280643296, 8.189428021541295488657516650099, 8.667376364818201047649156471127, 9.313046987954428372891966490218, 9.959403676084569457304383592453

Graph of the ZZ-function along the critical line