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Properties

Label 2-728-728.349-c1-0-93
Degree 22
Conductor 728728
Sign −0.860+0.508i-0.860 + 0.508i
Analytic cond. 5.813105.81310
Root an. cond. 2.411032.41103
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  + (−0.366 + 1.36i)2-s + (−0.636 − 1.10i)3-s + (−1.73 − i)4-s + (−0.465 − 0.465i)5-s + (1.73 − 0.465i)6-s + (−0.684 − 2.55i)7-s + (2 − 1.99i)8-s + (0.689 − 1.19i)9-s + (0.807 − 0.465i)10-s + 2.54i·12-s + (−0.102 + 3.60i)13-s + 3.74·14-s + (−0.217 + 0.810i)15-s + (1.99 + 3.46i)16-s + (1.37 + 1.37i)18-s + (−7.36 + 1.97i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.367 − 0.636i)3-s + (−0.866 − 0.5i)4-s + (−0.208 − 0.208i)5-s + (0.709 − 0.190i)6-s + (−0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (0.229 − 0.398i)9-s + (0.255 − 0.147i)10-s + 0.735i·12-s + (−0.0284 + 0.999i)13-s + 0.999·14-s + (−0.0560 + 0.209i)15-s + (0.499 + 0.866i)16-s + (0.325 + 0.325i)18-s + (−1.69 + 0.452i)19-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.860+0.508i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((−0.860+0.508i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.860+0.508i-0.860 + 0.508i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(349,⋅)\chi_{728} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :1/2), −0.860+0.508i)(2,\ 728,\ (\ :1/2),\ -0.860 + 0.508i)

Particular Values

L(1)L(1) ≈\approx 0.0652441−0.238624i0.0652441 - 0.238624i
L(12)L(\frac12) ≈\approx 0.0652441−0.238624i0.0652441 - 0.238624i
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+(0.366−1.36i)T 1 + (0.366 - 1.36i)T
7 1+(0.684+2.55i)T 1 + (0.684 + 2.55i)T
13 1+(0.102−3.60i)T 1 + (0.102 - 3.60i)T
good3 1+(0.636+1.10i)T+(−1.5+2.59i)T2 1 + (0.636 + 1.10i)T + (-1.5 + 2.59i)T^{2}
5 1+(0.465+0.465i)T+5iT2 1 + (0.465 + 0.465i)T + 5iT^{2}
11 1+(−9.52−5.5i)T2 1 + (-9.52 - 5.5i)T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(7.36−1.97i)T+(16.4−9.5i)T2 1 + (7.36 - 1.97i)T + (16.4 - 9.5i)T^{2}
23 1+(3.01−1.74i)T+(11.5−19.9i)T2 1 + (3.01 - 1.74i)T + (11.5 - 19.9i)T^{2}
29 1+(14.5−25.1i)T2 1 + (14.5 - 25.1i)T^{2}
31 1−31iT2 1 - 31iT^{2}
37 1+(−32.0−18.5i)T2 1 + (-32.0 - 18.5i)T^{2}
41 1+(35.5+20.5i)T2 1 + (35.5 + 20.5i)T^{2}
43 1+(−21.5−37.2i)T2 1 + (-21.5 - 37.2i)T^{2}
47 1+47iT2 1 + 47iT^{2}
53 1−53T2 1 - 53T^{2}
59 1+(3.35+12.5i)T+(−51.0+29.5i)T2 1 + (3.35 + 12.5i)T + (-51.0 + 29.5i)T^{2}
61 1+(7.77−13.4i)T+(−30.5−52.8i)T2 1 + (7.77 - 13.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(58.0+33.5i)T2 1 + (58.0 + 33.5i)T^{2}
71 1+(16.0−4.29i)T+(61.4−35.5i)T2 1 + (16.0 - 4.29i)T + (61.4 - 35.5i)T^{2}
73 1+73iT2 1 + 73iT^{2}
79 1−8.25T+79T2 1 - 8.25T + 79T^{2}
83 1+(4.00+4.00i)T+83iT2 1 + (4.00 + 4.00i)T + 83iT^{2}
89 1+(−77.0−44.5i)T2 1 + (-77.0 - 44.5i)T^{2}
97 1+(−84.0+48.5i)T2 1 + (-84.0 + 48.5i)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

−9.942089036439574040466107846365, −9.032654152181544780501766919342, −8.112224155332837948101903068761, −7.28908716189033445666212025116, −6.55183322714067573959079014489, −6.02616119867159604719137113636, −4.49346936447538378198986905397, −3.93845226643998305230010784173, −1.59435055305456697360215051448, −0.14668513627062788227937934034, 2.05085024351402051369105920173, 3.11311032912301199113347918327, 4.25420568744673253008654157702, 5.12104705048397543861131495207, 6.09063467451888288714503670939, 7.56517274192053396122705369945, 8.446301769361946638590173723833, 9.214931926902479206614245226792, 10.12713933662446218119771251781, 10.70712324460709729705977220793

Graph of the ZZ-function along the critical line