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Properties

Label 2-1638-13.10-c1-0-1
Degree 22
Conductor 16381638
Sign −0.636−0.770i-0.636 - 0.770i
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 − 3.38i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (1.69 + 2.93i)10-s + (−0.712 + 0.411i)11-s + (−2.74 + 2.33i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−2.29 + 3.96i)17-s + (−5.11 − 2.95i)19-s + (−2.93 − 1.69i)20-s + (0.411 − 0.712i)22-s + (3.06 + 5.30i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.51i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.535 + 0.928i)10-s + (−0.214 + 0.124i)11-s + (−0.762 + 0.646i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.555 + 0.962i)17-s + (−1.17 − 0.677i)19-s + (−0.656 − 0.378i)20-s + (0.0877 − 0.151i)22-s + (0.639 + 1.10i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((−0.636−0.770i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((−0.636−0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.636−0.770i-0.636 - 0.770i
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), −0.636−0.770i)(2,\ 1638,\ (\ :1/2),\ -0.636 - 0.770i)

Particular Values

L(1)L(1) ≈\approx 0.34898018370.3489801837
L(12)L(\frac12) ≈\approx 0.34898018370.3489801837
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+(2.74−2.33i)T 1 + (2.74 - 2.33i)T
good5 1+3.38iT−5T2 1 + 3.38iT - 5T^{2}
11 1+(0.712−0.411i)T+(5.5−9.52i)T2 1 + (0.712 - 0.411i)T + (5.5 - 9.52i)T^{2}
17 1+(2.29−3.96i)T+(−8.5−14.7i)T2 1 + (2.29 - 3.96i)T + (-8.5 - 14.7i)T^{2}
19 1+(5.11+2.95i)T+(9.5+16.4i)T2 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2}
23 1+(−3.06−5.30i)T+(−11.5+19.9i)T2 1 + (-3.06 - 5.30i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.43+5.94i)T+(−14.5+25.1i)T2 1 + (3.43 + 5.94i)T + (-14.5 + 25.1i)T^{2}
31 1−4.28iT−31T2 1 - 4.28iT - 31T^{2}
37 1+(8.39−4.84i)T+(18.5−32.0i)T2 1 + (8.39 - 4.84i)T + (18.5 - 32.0i)T^{2}
41 1+(−0.0774+0.0446i)T+(20.5−35.5i)T2 1 + (-0.0774 + 0.0446i)T + (20.5 - 35.5i)T^{2}
43 1+(−3.67+6.36i)T+(−21.5−37.2i)T2 1 + (-3.67 + 6.36i)T + (-21.5 - 37.2i)T^{2}
47 1−11.1iT−47T2 1 - 11.1iT - 47T^{2}
53 1−7.01T+53T2 1 - 7.01T + 53T^{2}
59 1+(1.50+0.870i)T+(29.5+51.0i)T2 1 + (1.50 + 0.870i)T + (29.5 + 51.0i)T^{2}
61 1+(1.18−2.06i)T+(−30.5−52.8i)T2 1 + (1.18 - 2.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.252+0.145i)T+(33.5−58.0i)T2 1 + (-0.252 + 0.145i)T + (33.5 - 58.0i)T^{2}
71 1+(9.48+5.47i)T+(35.5+61.4i)T2 1 + (9.48 + 5.47i)T + (35.5 + 61.4i)T^{2}
73 1−12.7iT−73T2 1 - 12.7iT - 73T^{2}
79 1−9.95T+79T2 1 - 9.95T + 79T^{2}
83 1−3.23iT−83T2 1 - 3.23iT - 83T^{2}
89 1+(6.96−4.02i)T+(44.5−77.0i)T2 1 + (6.96 - 4.02i)T + (44.5 - 77.0i)T^{2}
97 1+(−12.7−7.38i)T+(48.5+84.0i)T2 1 + (-12.7 - 7.38i)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

−9.292806056201695341897659910878, −8.909919420582610003889286675684, −8.269808848678991394403408977170, −7.44063697021978196097865065754, −6.53710817701216606885653291950, −5.51722177001712952959365065906, −4.83718730072515938933287147319, −4.07724472570487124447950129454, −2.28012032687777358265253611413, −1.37158764093879750724689664774, 0.16023752689218488783500413660, 2.09592991444404758346045938817, 2.78713441193942321906303362060, 3.72506308943262425294860255855, 4.90507197307536261396958564945, 6.06486770200072431270940743928, 7.03573285839534686488053694154, 7.34079684565019894387880705732, 8.335642753472153944179758535003, 9.128687651334172995174709709578

Graph of the ZZ-function along the critical line