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Properties

Label 2-980-140.67-c1-0-29
Degree 22
Conductor 980980
Sign −0.593−0.805i-0.593 - 0.805i
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  + (0.805 + 1.16i)2-s + (−3.15 + 0.844i)3-s + (−0.703 + 1.87i)4-s + (−0.260 − 2.22i)5-s + (−3.52 − 2.98i)6-s + (−2.74 + 0.689i)8-s + (6.62 − 3.82i)9-s + (2.37 − 2.09i)10-s + (1.96 + 1.13i)11-s + (0.635 − 6.49i)12-s + (1.38 − 1.38i)13-s + (2.69 + 6.78i)15-s + (−3.01 − 2.63i)16-s + (−0.186 + 0.0499i)17-s + (9.78 + 4.62i)18-s + (3.45 + 5.98i)19-s + ⋯
L(s)  = 1  + (0.569 + 0.822i)2-s + (−1.81 + 0.487i)3-s + (−0.351 + 0.936i)4-s + (−0.116 − 0.993i)5-s + (−1.43 − 1.21i)6-s + (−0.969 + 0.243i)8-s + (2.20 − 1.27i)9-s + (0.750 − 0.661i)10-s + (0.593 + 0.342i)11-s + (0.183 − 1.87i)12-s + (0.383 − 0.383i)13-s + (0.696 + 1.75i)15-s + (−0.752 − 0.658i)16-s + (−0.0451 + 0.0121i)17-s + (2.30 + 1.08i)18-s + (0.792 + 1.37i)19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.593−0.805i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.593−0.805i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.593−0.805i-0.593 - 0.805i
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), −0.593−0.805i)(2,\ 980,\ (\ :1/2),\ -0.593 - 0.805i)

Particular Values

L(1)L(1) ≈\approx 0.434269+0.859375i0.434269 + 0.859375i
L(12)L(\frac12) ≈\approx 0.434269+0.859375i0.434269 + 0.859375i
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+(−0.805−1.16i)T 1 + (-0.805 - 1.16i)T
5 1+(0.260+2.22i)T 1 + (0.260 + 2.22i)T
7 1 1
good3 1+(3.15−0.844i)T+(2.59−1.5i)T2 1 + (3.15 - 0.844i)T + (2.59 - 1.5i)T^{2}
11 1+(−1.96−1.13i)T+(5.5+9.52i)T2 1 + (-1.96 - 1.13i)T + (5.5 + 9.52i)T^{2}
13 1+(−1.38+1.38i)T−13iT2 1 + (-1.38 + 1.38i)T - 13iT^{2}
17 1+(0.186−0.0499i)T+(14.7−8.5i)T2 1 + (0.186 - 0.0499i)T + (14.7 - 8.5i)T^{2}
19 1+(−3.45−5.98i)T+(−9.5+16.4i)T2 1 + (-3.45 - 5.98i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.866+3.23i)T+(−19.9−11.5i)T2 1 + (-0.866 + 3.23i)T + (-19.9 - 11.5i)T^{2}
29 1−7.33iT−29T2 1 - 7.33iT - 29T^{2}
31 1+(0.430+0.248i)T+(15.5+26.8i)T2 1 + (0.430 + 0.248i)T + (15.5 + 26.8i)T^{2}
37 1+(−0.904+3.37i)T+(−32.0−18.5i)T2 1 + (-0.904 + 3.37i)T + (-32.0 - 18.5i)T^{2}
41 1−3.22T+41T2 1 - 3.22T + 41T^{2}
43 1+(2.91+2.91i)T+43iT2 1 + (2.91 + 2.91i)T + 43iT^{2}
47 1+(2.40+0.645i)T+(40.7+23.5i)T2 1 + (2.40 + 0.645i)T + (40.7 + 23.5i)T^{2}
53 1+(−1.87−6.98i)T+(−45.8+26.5i)T2 1 + (-1.87 - 6.98i)T + (-45.8 + 26.5i)T^{2}
59 1+(2.61−4.52i)T+(−29.5−51.0i)T2 1 + (2.61 - 4.52i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.00−8.67i)T+(−30.5+52.8i)T2 1 + (-5.00 - 8.67i)T + (-30.5 + 52.8i)T^{2}
67 1+(−3.21−12.0i)T+(−58.0+33.5i)T2 1 + (-3.21 - 12.0i)T + (-58.0 + 33.5i)T^{2}
71 1+3.60iT−71T2 1 + 3.60iT - 71T^{2}
73 1+(−3.38−12.6i)T+(−63.2+36.5i)T2 1 + (-3.38 - 12.6i)T + (-63.2 + 36.5i)T^{2}
79 1+(−5.66−9.81i)T+(−39.5+68.4i)T2 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.591+0.591i)T+83iT2 1 + (0.591 + 0.591i)T + 83iT^{2}
89 1+(−3.45+1.99i)T+(44.5−77.0i)T2 1 + (-3.45 + 1.99i)T + (44.5 - 77.0i)T^{2}
97 1+(1.09+1.09i)T+97iT2 1 + (1.09 + 1.09i)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

−10.32922785023187349981177882764, −9.472679660891283304745173179995, −8.598029420348758065872538991256, −7.47010874799429363914405338068, −6.63808285461829269090794026788, −5.69173833362550135702731362216, −5.34418952417490404518538677654, −4.38000139879729101727031468706, −3.74835381928925044828831688422, −1.09404189236223748914786342805, 0.59182551506647517843409121115, 1.91983314126946747215255612995, 3.37951685175502599305870706881, 4.50547089033427457514743061719, 5.35008927866693357068054999790, 6.35342114077628943887658421509, 6.60160024390806552192115268658, 7.69897678285084717910774328245, 9.366219766907498954342111316328, 10.06991602622884063160034071122

Graph of the ZZ-function along the critical line