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Properties

Label 2-1638-13.10-c1-0-21
Degree 22
Conductor 16381638
Sign 0.545+0.838i0.545 + 0.838i
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 + 1.56i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.781 + 1.35i)10-s + (2.48 − 1.43i)11-s + (2.99 − 2.00i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.11 − 1.93i)17-s + (−6.26 − 3.61i)19-s + (1.35 + 0.781i)20-s + (1.43 − 2.48i)22-s + (0.833 + 1.44i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.699i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.247 + 0.428i)10-s + (0.748 − 0.432i)11-s + (0.830 − 0.556i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.270 − 0.468i)17-s + (−1.43 − 0.830i)19-s + (0.302 + 0.174i)20-s + (0.305 − 0.529i)22-s + (0.173 + 0.301i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.545+0.838i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.545+0.838i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.545+0.838i0.545 + 0.838i
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), 0.545+0.838i)(2,\ 1638,\ (\ :1/2),\ 0.545 + 0.838i)

Particular Values

L(1)L(1) ≈\approx 2.4830040352.483004035
L(12)L(\frac12) ≈\approx 2.4830040352.483004035
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+(−2.99+2.00i)T 1 + (-2.99 + 2.00i)T
good5 1−1.56iT−5T2 1 - 1.56iT - 5T^{2}
11 1+(−2.48+1.43i)T+(5.5−9.52i)T2 1 + (-2.48 + 1.43i)T + (5.5 - 9.52i)T^{2}
17 1+(−1.11+1.93i)T+(−8.5−14.7i)T2 1 + (-1.11 + 1.93i)T + (-8.5 - 14.7i)T^{2}
19 1+(6.26+3.61i)T+(9.5+16.4i)T2 1 + (6.26 + 3.61i)T + (9.5 + 16.4i)T^{2}
23 1+(−0.833−1.44i)T+(−11.5+19.9i)T2 1 + (-0.833 - 1.44i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.41−4.18i)T+(−14.5+25.1i)T2 1 + (-2.41 - 4.18i)T + (-14.5 + 25.1i)T^{2}
31 1−0.597iT−31T2 1 - 0.597iT - 31T^{2}
37 1+(−0.0333+0.0192i)T+(18.5−32.0i)T2 1 + (-0.0333 + 0.0192i)T + (18.5 - 32.0i)T^{2}
41 1+(−6.88+3.97i)T+(20.5−35.5i)T2 1 + (-6.88 + 3.97i)T + (20.5 - 35.5i)T^{2}
43 1+(−5.04+8.73i)T+(−21.5−37.2i)T2 1 + (-5.04 + 8.73i)T + (-21.5 - 37.2i)T^{2}
47 1−7.02iT−47T2 1 - 7.02iT - 47T^{2}
53 1−5.98T+53T2 1 - 5.98T + 53T^{2}
59 1+(0.776+0.448i)T+(29.5+51.0i)T2 1 + (0.776 + 0.448i)T + (29.5 + 51.0i)T^{2}
61 1+(−7.12+12.3i)T+(−30.5−52.8i)T2 1 + (-7.12 + 12.3i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.42−0.820i)T+(33.5−58.0i)T2 1 + (1.42 - 0.820i)T + (33.5 - 58.0i)T^{2}
71 1+(−1.98−1.14i)T+(35.5+61.4i)T2 1 + (-1.98 - 1.14i)T + (35.5 + 61.4i)T^{2}
73 1+11.2iT−73T2 1 + 11.2iT - 73T^{2}
79 1−4.26T+79T2 1 - 4.26T + 79T^{2}
83 1+4.94iT−83T2 1 + 4.94iT - 83T^{2}
89 1+(2.09−1.21i)T+(44.5−77.0i)T2 1 + (2.09 - 1.21i)T + (44.5 - 77.0i)T^{2}
97 1+(4.23+2.44i)T+(48.5+84.0i)T2 1 + (4.23 + 2.44i)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

−9.229709581906116679830668668932, −8.649430865346620060428415471582, −7.43497358279463259086057394714, −6.64082985295823651550907442276, −6.11770329685873681563835628279, −5.08985064098628792087465970329, −4.00563555074576647604197488362, −3.30308087923947761213260794989, −2.39991897578885041132813792532, −0.884122202246548482612109342596, 1.32438888429897324348160615643, 2.57009290462463113762655907093, 4.05969386808096150298022597000, 4.23085033681988316625214423131, 5.51042994829537604547785030778, 6.27608401239095868569491536960, 6.81394036175540004407261590616, 8.037356086098102184663701297798, 8.607544936891610342749944323932, 9.325682861885630036064654778026

Graph of the ZZ-function along the critical line