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Properties

Label 2-80-16.5-c1-0-6
Degree 22
Conductor 8080
Sign 0.321+0.946i0.321 + 0.946i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
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  + (0.562 − 1.29i)2-s + (0.209 − 0.209i)3-s + (−1.36 − 1.45i)4-s + (0.707 + 0.707i)5-s + (−0.154 − 0.389i)6-s − 1.73i·7-s + (−2.66 + 0.952i)8-s + 2.91i·9-s + (1.31 − 0.519i)10-s + (0.505 + 0.505i)11-s + (−0.592 − 0.0194i)12-s + (−1.88 + 1.88i)13-s + (−2.25 − 0.977i)14-s + 0.296·15-s + (−0.262 + 3.99i)16-s + 4.53·17-s + ⋯
L(s)  = 1  + (0.397 − 0.917i)2-s + (0.120 − 0.120i)3-s + (−0.683 − 0.729i)4-s + (0.316 + 0.316i)5-s + (−0.0628 − 0.159i)6-s − 0.656i·7-s + (−0.941 + 0.336i)8-s + 0.970i·9-s + (0.415 − 0.164i)10-s + (0.152 + 0.152i)11-s + (−0.171 − 0.00561i)12-s + (−0.523 + 0.523i)13-s + (−0.602 − 0.261i)14-s + 0.0765·15-s + (−0.0655 + 0.997i)16-s + 1.09·17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.321+0.946i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.321+0.946i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.321+0.946i0.321 + 0.946i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(21,⋅)\chi_{80} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.321+0.946i)(2,\ 80,\ (\ :1/2),\ 0.321 + 0.946i)

Particular Values

L(1)L(1) ≈\approx 0.910638−0.652648i0.910638 - 0.652648i
L(12)L(\frac12) ≈\approx 0.910638−0.652648i0.910638 - 0.652648i
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+(−0.562+1.29i)T 1 + (-0.562 + 1.29i)T
5 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good3 1+(−0.209+0.209i)T−3iT2 1 + (-0.209 + 0.209i)T - 3iT^{2}
7 1+1.73iT−7T2 1 + 1.73iT - 7T^{2}
11 1+(−0.505−0.505i)T+11iT2 1 + (-0.505 - 0.505i)T + 11iT^{2}
13 1+(1.88−1.88i)T−13iT2 1 + (1.88 - 1.88i)T - 13iT^{2}
17 1−4.53T+17T2 1 - 4.53T + 17T^{2}
19 1+(3.22−3.22i)T−19iT2 1 + (3.22 - 3.22i)T - 19iT^{2}
23 1+8.85iT−23T2 1 + 8.85iT - 23T^{2}
29 1+(2.44−2.44i)T−29iT2 1 + (2.44 - 2.44i)T - 29iT^{2}
31 1+5.70T+31T2 1 + 5.70T + 31T^{2}
37 1+(5.35+5.35i)T+37iT2 1 + (5.35 + 5.35i)T + 37iT^{2}
41 1+10.0iT−41T2 1 + 10.0iT - 41T^{2}
43 1+(2.10+2.10i)T+43iT2 1 + (2.10 + 2.10i)T + 43iT^{2}
47 1−4.32T+47T2 1 - 4.32T + 47T^{2}
53 1+(1.37+1.37i)T+53iT2 1 + (1.37 + 1.37i)T + 53iT^{2}
59 1+(−6.64−6.64i)T+59iT2 1 + (-6.64 - 6.64i)T + 59iT^{2}
61 1+(−5.26+5.26i)T−61iT2 1 + (-5.26 + 5.26i)T - 61iT^{2}
67 1+(10.5−10.5i)T−67iT2 1 + (10.5 - 10.5i)T - 67iT^{2}
71 1−14.0iT−71T2 1 - 14.0iT - 71T^{2}
73 1−6.63iT−73T2 1 - 6.63iT - 73T^{2}
79 1−4.27T+79T2 1 - 4.27T + 79T^{2}
83 1+(−9.15+9.15i)T−83iT2 1 + (-9.15 + 9.15i)T - 83iT^{2}
89 1+3.23iT−89T2 1 + 3.23iT - 89T^{2}
97 1−1.94T+97T2 1 - 1.94T + 97T^{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

−14.27371398753111586618978286863, −13.03219913014797193643474418612, −12.14337761863172029024201584505, −10.68555411513024738462666858584, −10.23417910637396873061323568057, −8.743489391040442548666316258199, −7.15866525257726208591423397397, −5.46835453361246655882758634900, −4.00474697638882823292828938126, −2.15694204486794891944584670387, 3.40262913038877110843840701119, 5.16220912860181285299764065021, 6.19630405849260286173486799408, 7.64420686629694987094620622626, 8.935514093002883866758704993845, 9.719099882024863277772482871803, 11.76233078115189870350164928693, 12.63523425561765253982061124271, 13.64836172894177716997041997252, 14.92799911556472090816870719680

Graph of the ZZ-function along the critical line