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Properties

Label 2-1638-13.10-c1-0-19
Degree 22
Conductor 16381638
Sign 0.996−0.0797i0.996 - 0.0797i
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 3.18i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (1.59 + 2.75i)10-s + (3.49 − 2.01i)11-s + (−0.333 − 3.59i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (1.23 − 2.13i)17-s + (0.595 + 0.343i)19-s + (2.75 + 1.59i)20-s + (2.01 − 3.49i)22-s + (2.89 + 5.01i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 1.42i·5-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.503 + 0.871i)10-s + (1.05 − 0.608i)11-s + (−0.0925 − 0.995i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.298 − 0.517i)17-s + (0.136 + 0.0788i)19-s + (0.616 + 0.355i)20-s + (0.429 − 0.744i)22-s + (0.603 + 1.04i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.996−0.0797i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0797i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.996−0.0797i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0797i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.996−0.0797i0.996 - 0.0797i
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1638(127,⋅)\chi_{1638} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 0.996−0.0797i)(2,\ 1638,\ (\ :1/2),\ 0.996 - 0.0797i)

Particular Values

L(1)L(1) ≈\approx 2.7763934052.776393405
L(12)L(\frac12) ≈\approx 2.7763934052.776393405
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1 1
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(0.333+3.59i)T 1 + (0.333 + 3.59i)T
good5 1−3.18iT−5T2 1 - 3.18iT - 5T^{2}
11 1+(−3.49+2.01i)T+(5.5−9.52i)T2 1 + (-3.49 + 2.01i)T + (5.5 - 9.52i)T^{2}
17 1+(−1.23+2.13i)T+(−8.5−14.7i)T2 1 + (-1.23 + 2.13i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.595−0.343i)T+(9.5+16.4i)T2 1 + (-0.595 - 0.343i)T + (9.5 + 16.4i)T^{2}
23 1+(−2.89−5.01i)T+(−11.5+19.9i)T2 1 + (-2.89 - 5.01i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.940−1.62i)T+(−14.5+25.1i)T2 1 + (-0.940 - 1.62i)T + (-14.5 + 25.1i)T^{2}
31 1−4.17iT−31T2 1 - 4.17iT - 31T^{2}
37 1+(0.322−0.186i)T+(18.5−32.0i)T2 1 + (0.322 - 0.186i)T + (18.5 - 32.0i)T^{2}
41 1+(−6.83+3.94i)T+(20.5−35.5i)T2 1 + (-6.83 + 3.94i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.48+4.30i)T+(−21.5−37.2i)T2 1 + (-2.48 + 4.30i)T + (-21.5 - 37.2i)T^{2}
47 1−11.8iT−47T2 1 - 11.8iT - 47T^{2}
53 1−9.59T+53T2 1 - 9.59T + 53T^{2}
59 1+(−2.23−1.28i)T+(29.5+51.0i)T2 1 + (-2.23 - 1.28i)T + (29.5 + 51.0i)T^{2}
61 1+(6.79−11.7i)T+(−30.5−52.8i)T2 1 + (6.79 - 11.7i)T + (-30.5 - 52.8i)T^{2}
67 1+(−7.46+4.30i)T+(33.5−58.0i)T2 1 + (-7.46 + 4.30i)T + (33.5 - 58.0i)T^{2}
71 1+(9.08+5.24i)T+(35.5+61.4i)T2 1 + (9.08 + 5.24i)T + (35.5 + 61.4i)T^{2}
73 1−16.5iT−73T2 1 - 16.5iT - 73T^{2}
79 1+13.2T+79T2 1 + 13.2T + 79T^{2}
83 1+9.39iT−83T2 1 + 9.39iT - 83T^{2}
89 1+(−7.55+4.36i)T+(44.5−77.0i)T2 1 + (-7.55 + 4.36i)T + (44.5 - 77.0i)T^{2}
97 1+(5.32+3.07i)T+(48.5+84.0i)T2 1 + (5.32 + 3.07i)T + (48.5 + 84.0i)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

−9.542151954648198413821483467127, −8.687364622808177416767319622277, −7.47427107870414629557402156905, −7.02166407489691021968660557827, −5.98113535034032303715034547849, −5.45644431816364556029486624353, −4.17108696034467267694017045193, −3.22598257254841351757621455416, −2.72639865812474435270555542153, −1.23604589551708631757880039394, 1.10004135052111374271776522232, 2.19634750366983714545599702794, 3.87236000021475097725937572961, 4.43221314079495573877756051133, 5.05706809357876312300273059686, 6.10191277539369852629920919059, 6.86037232567704664410831981590, 7.76890683837375474907772867511, 8.628335844473711346460086978250, 9.151350746795450580062187841449

Graph of the ZZ-function along the critical line