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Properties

Label 2-2e4-16.13-c5-0-2
Degree 22
Conductor 1616
Sign 0.979+0.201i0.979 + 0.201i
Analytic cond. 2.566142.56614
Root an. cond. 1.601911.60191
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.36 − 4.54i)2-s + (16.8 + 16.8i)3-s + (−9.38 + 30.5i)4-s + (66.0 − 66.0i)5-s + (20.0 − 133. i)6-s + 75.3i·7-s + (170. − 60.2i)8-s + 327. i·9-s + (−522. − 78.2i)10-s + (−79.0 + 79.0i)11-s + (−675. + 358. i)12-s + (−238. − 238. i)13-s + (342. − 253. i)14-s + 2.23e3·15-s + (−847. − 574. i)16-s − 1.75e3·17-s + ⋯
L(s)  = 1  + (−0.594 − 0.804i)2-s + (1.08 + 1.08i)3-s + (−0.293 + 0.956i)4-s + (1.18 − 1.18i)5-s + (0.227 − 1.51i)6-s + 0.580i·7-s + (0.943 − 0.332i)8-s + 1.34i·9-s + (−1.65 − 0.247i)10-s + (−0.197 + 0.197i)11-s + (−1.35 + 0.718i)12-s + (−0.391 − 0.391i)13-s + (0.467 − 0.345i)14-s + 2.55·15-s + (−0.828 − 0.560i)16-s − 1.47·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.979+0.201i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+5/2) L(s)=((0.979+0.201i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.979+0.201i0.979 + 0.201i
Analytic conductor: 2.566142.56614
Root analytic conductor: 1.601911.60191
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ16(13,⋅)\chi_{16} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 16, ( :5/2), 0.979+0.201i)(2,\ 16,\ (\ :5/2),\ 0.979 + 0.201i)

Particular Values

L(3)L(3) ≈\approx 1.45190−0.147426i1.45190 - 0.147426i
L(12)L(\frac12) ≈\approx 1.45190−0.147426i1.45190 - 0.147426i
L(72)L(\frac{7}{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+(3.36+4.54i)T 1 + (3.36 + 4.54i)T
good3 1+(−16.8−16.8i)T+243iT2 1 + (-16.8 - 16.8i)T + 243iT^{2}
5 1+(−66.0+66.0i)T−3.12e3iT2 1 + (-66.0 + 66.0i)T - 3.12e3iT^{2}
7 1−75.3iT−1.68e4T2 1 - 75.3iT - 1.68e4T^{2}
11 1+(79.0−79.0i)T−1.61e5iT2 1 + (79.0 - 79.0i)T - 1.61e5iT^{2}
13 1+(238.+238.i)T+3.71e5iT2 1 + (238. + 238. i)T + 3.71e5iT^{2}
17 1+1.75e3T+1.41e6T2 1 + 1.75e3T + 1.41e6T^{2}
19 1+(311.+311.i)T+2.47e6iT2 1 + (311. + 311. i)T + 2.47e6iT^{2}
23 1+1.30e3iT−6.43e6T2 1 + 1.30e3iT - 6.43e6T^{2}
29 1+(−2.58e3−2.58e3i)T+2.05e7iT2 1 + (-2.58e3 - 2.58e3i)T + 2.05e7iT^{2}
31 1+2.37e3T+2.86e7T2 1 + 2.37e3T + 2.86e7T^{2}
37 1+(1.94e3−1.94e3i)T−6.93e7iT2 1 + (1.94e3 - 1.94e3i)T - 6.93e7iT^{2}
41 1+3.91e3iT−1.15e8T2 1 + 3.91e3iT - 1.15e8T^{2}
43 1+(8.82e3−8.82e3i)T−1.47e8iT2 1 + (8.82e3 - 8.82e3i)T - 1.47e8iT^{2}
47 1−2.32e4T+2.29e8T2 1 - 2.32e4T + 2.29e8T^{2}
53 1+(−7.57e3+7.57e3i)T−4.18e8iT2 1 + (-7.57e3 + 7.57e3i)T - 4.18e8iT^{2}
59 1+(3.30e4−3.30e4i)T−7.14e8iT2 1 + (3.30e4 - 3.30e4i)T - 7.14e8iT^{2}
61 1+(2.65e4+2.65e4i)T+8.44e8iT2 1 + (2.65e4 + 2.65e4i)T + 8.44e8iT^{2}
67 1+(−1.62e4−1.62e4i)T+1.35e9iT2 1 + (-1.62e4 - 1.62e4i)T + 1.35e9iT^{2}
71 1+3.36e4iT−1.80e9T2 1 + 3.36e4iT - 1.80e9T^{2}
73 1−5.62e4iT−2.07e9T2 1 - 5.62e4iT - 2.07e9T^{2}
79 1−1.29e3T+3.07e9T2 1 - 1.29e3T + 3.07e9T^{2}
83 1+(−7.87e4−7.87e4i)T+3.93e9iT2 1 + (-7.87e4 - 7.87e4i)T + 3.93e9iT^{2}
89 1+8.69e4iT−5.58e9T2 1 + 8.69e4iT - 5.58e9T^{2}
97 1+8.39e3T+8.58e9T2 1 + 8.39e3T + 8.58e9T^{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

−18.04183857077556338942310878652, −16.83934378257137916947476103435, −15.52310235200364162020144960951, −13.76024669780051736972001380645, −12.60974939448344019752422483101, −10.39214730947159230854465824278, −9.231705832244961714987618988788, −8.630642809896823925426925905941, −4.69951836598634660717162508635, −2.37083630931323561727197754596, 2.11562578316265800102810159623, 6.44121087447635631979554002876, 7.41805753849027211773815548850, 9.099721454339143765453168463615, 10.56937935045767034382067538782, 13.51969117677511667442166438823, 13.96680686787847730723653240590, 15.13610460839243207197984041274, 17.24953405083272979785323212165, 18.17372333654813981943403970284

Graph of the ZZ-function along the critical line