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Properties

Label 2-980-140.67-c1-0-90
Degree 22
Conductor 980980
Sign 0.940−0.341i0.940 - 0.341i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (1.18 + 0.764i)2-s + (2.38 − 0.638i)3-s + (0.831 + 1.81i)4-s + (0.525 − 2.17i)5-s + (3.32 + 1.06i)6-s + (−0.401 + 2.79i)8-s + (2.68 − 1.54i)9-s + (2.28 − 2.18i)10-s + (4.09 + 2.36i)11-s + (3.14 + 3.80i)12-s + (−0.0592 + 0.0592i)13-s + (−0.135 − 5.51i)15-s + (−2.61 + 3.02i)16-s + (−4.77 + 1.27i)17-s + (4.37 + 0.207i)18-s + (−1.31 − 2.27i)19-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (1.37 − 0.368i)3-s + (0.415 + 0.909i)4-s + (0.235 − 0.971i)5-s + (1.35 + 0.433i)6-s + (−0.142 + 0.989i)8-s + (0.893 − 0.515i)9-s + (0.723 − 0.690i)10-s + (1.23 + 0.713i)11-s + (0.907 + 1.09i)12-s + (−0.0164 + 0.0164i)13-s + (−0.0349 − 1.42i)15-s + (−0.654 + 0.755i)16-s + (−1.15 + 0.310i)17-s + (1.03 + 0.0490i)18-s + (−0.301 − 0.522i)19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.940−0.341i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.940−0.341i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.940−0.341i0.940 - 0.341i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(67,⋅)\chi_{980} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.940−0.341i)(2,\ 980,\ (\ :1/2),\ 0.940 - 0.341i)

Particular Values

L(1)L(1) ≈\approx 4.12998+0.725974i4.12998 + 0.725974i
L(12)L(\frac12) ≈\approx 4.12998+0.725974i4.12998 + 0.725974i
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+(−1.18−0.764i)T 1 + (-1.18 - 0.764i)T
5 1+(−0.525+2.17i)T 1 + (-0.525 + 2.17i)T
7 1 1
good3 1+(−2.38+0.638i)T+(2.59−1.5i)T2 1 + (-2.38 + 0.638i)T + (2.59 - 1.5i)T^{2}
11 1+(−4.09−2.36i)T+(5.5+9.52i)T2 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2}
13 1+(0.0592−0.0592i)T−13iT2 1 + (0.0592 - 0.0592i)T - 13iT^{2}
17 1+(4.77−1.27i)T+(14.7−8.5i)T2 1 + (4.77 - 1.27i)T + (14.7 - 8.5i)T^{2}
19 1+(1.31+2.27i)T+(−9.5+16.4i)T2 1 + (1.31 + 2.27i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.292+1.09i)T+(−19.9−11.5i)T2 1 + (-0.292 + 1.09i)T + (-19.9 - 11.5i)T^{2}
29 1+7.27iT−29T2 1 + 7.27iT - 29T^{2}
31 1+(4.01+2.31i)T+(15.5+26.8i)T2 1 + (4.01 + 2.31i)T + (15.5 + 26.8i)T^{2}
37 1+(0.596−2.22i)T+(−32.0−18.5i)T2 1 + (0.596 - 2.22i)T + (-32.0 - 18.5i)T^{2}
41 1−5.71T+41T2 1 - 5.71T + 41T^{2}
43 1+(−1.57−1.57i)T+43iT2 1 + (-1.57 - 1.57i)T + 43iT^{2}
47 1+(−2.67−0.716i)T+(40.7+23.5i)T2 1 + (-2.67 - 0.716i)T + (40.7 + 23.5i)T^{2}
53 1+(−2.44−9.12i)T+(−45.8+26.5i)T2 1 + (-2.44 - 9.12i)T + (-45.8 + 26.5i)T^{2}
59 1+(1.67−2.90i)T+(−29.5−51.0i)T2 1 + (1.67 - 2.90i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.978−1.69i)T+(−30.5+52.8i)T2 1 + (-0.978 - 1.69i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.131−0.491i)T+(−58.0+33.5i)T2 1 + (-0.131 - 0.491i)T + (-58.0 + 33.5i)T^{2}
71 1+14.4iT−71T2 1 + 14.4iT - 71T^{2}
73 1+(−2.48−9.26i)T+(−63.2+36.5i)T2 1 + (-2.48 - 9.26i)T + (-63.2 + 36.5i)T^{2}
79 1+(3.05+5.28i)T+(−39.5+68.4i)T2 1 + (3.05 + 5.28i)T + (-39.5 + 68.4i)T^{2}
83 1+(5.62+5.62i)T+83iT2 1 + (5.62 + 5.62i)T + 83iT^{2}
89 1+(14.4−8.34i)T+(44.5−77.0i)T2 1 + (14.4 - 8.34i)T + (44.5 - 77.0i)T^{2}
97 1+(5.81+5.81i)T+97iT2 1 + (5.81 + 5.81i)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

−9.518810159492641868291988055985, −9.016741958367439783287105333773, −8.366871266019776456130481734969, −7.53710511187554216348102906957, −6.71730848133527081854950392344, −5.80323644252566141412174236896, −4.38381417401974506317252798302, −4.10622933435354400686546640929, −2.63066145608185318987548801893, −1.77619043994412504878204182777, 1.77541353535031493278708692314, 2.74613947310342081047527848496, 3.57119413057612448528207724976, 4.13054311662669088717838080289, 5.55807062298236073060091378663, 6.57078454034709302488437154918, 7.21967007394398239982789581530, 8.581463214593794337122973925000, 9.230521678657086372665068385504, 9.934853955076152138727961486555

Graph of the ZZ-function along the critical line