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Properties

Label 2-3808-3808.2141-c0-0-6
Degree 22
Conductor 38083808
Sign 0.831−0.555i0.831 - 0.555i
Analytic cond. 1.900431.90043
Root an. cond. 1.378561.37856
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.156 + 0.987i)2-s + (0.431 + 0.178i)3-s + (−0.951 + 0.309i)4-s + (−0.0600 − 0.144i)5-s + (−0.108 + 0.453i)6-s + (0.707 − 0.707i)7-s + (−0.453 − 0.891i)8-s + (−0.552 − 0.552i)9-s + (0.133 − 0.0819i)10-s + (−0.465 − 0.0366i)12-s + (0.809 + 0.587i)14-s − 0.0732i·15-s + (0.809 − 0.587i)16-s + i·17-s + (0.459 − 0.632i)18-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.431 + 0.178i)3-s + (−0.951 + 0.309i)4-s + (−0.0600 − 0.144i)5-s + (−0.108 + 0.453i)6-s + (0.707 − 0.707i)7-s + (−0.453 − 0.891i)8-s + (−0.552 − 0.552i)9-s + (0.133 − 0.0819i)10-s + (−0.465 − 0.0366i)12-s + (0.809 + 0.587i)14-s − 0.0732i·15-s + (0.809 − 0.587i)16-s + i·17-s + (0.459 − 0.632i)18-s + ⋯

Functional equation

Λ(s)=(3808s/2 ΓC(s) L(s)=((0.831−0.555i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3808s/2 ΓC(s) L(s)=((0.831−0.555i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38083808    =    25⋅7⋅172^{5} \cdot 7 \cdot 17
Sign: 0.831−0.555i0.831 - 0.555i
Analytic conductor: 1.900431.90043
Root analytic conductor: 1.378561.37856
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3808(2141,⋅)\chi_{3808} (2141, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3808, ( :0), 0.831−0.555i)(2,\ 3808,\ (\ :0),\ 0.831 - 0.555i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4570662811.457066281
L(12)L(\frac12) ≈\approx 1.4570662811.457066281
L(1)L(1) 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+(−0.156−0.987i)T 1 + (-0.156 - 0.987i)T
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
17 1−iT 1 - iT
good3 1+(−0.431−0.178i)T+(0.707+0.707i)T2 1 + (-0.431 - 0.178i)T + (0.707 + 0.707i)T^{2}
5 1+(0.0600+0.144i)T+(−0.707+0.707i)T2 1 + (0.0600 + 0.144i)T + (-0.707 + 0.707i)T^{2}
11 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
13 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
19 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
23 1−iT2 1 - iT^{2}
29 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
31 1−0.618T+T2 1 - 0.618T + T^{2}
37 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
41 1+(1.14+1.14i)T+iT2 1 + (1.14 + 1.14i)T + iT^{2}
43 1+(−1.57+0.652i)T+(0.707−0.707i)T2 1 + (-1.57 + 0.652i)T + (0.707 - 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−1.40+0.581i)T+(0.707−0.707i)T2 1 + (-1.40 + 0.581i)T + (0.707 - 0.707i)T^{2}
59 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
61 1+(−1.79−0.744i)T+(0.707+0.707i)T2 1 + (-1.79 - 0.744i)T + (0.707 + 0.707i)T^{2}
67 1+(−1.20−0.497i)T+(0.707+0.707i)T2 1 + (-1.20 - 0.497i)T + (0.707 + 0.707i)T^{2}
71 1+iT2 1 + iT^{2}
73 1+(−0.221−0.221i)T+iT2 1 + (-0.221 - 0.221i)T + iT^{2}
79 1+T2 1 + T^{2}
83 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
89 1+iT2 1 + iT^{2}
97 1+0.907T+T2 1 + 0.907T + 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

−8.409394323907232438414931847669, −8.256584426313307907552888885685, −7.21447583489220768547407785535, −6.65833715354765458922270208680, −5.75172627033416960371049091879, −5.08134272166875999838634019705, −4.03595801478624944567091455210, −3.77601627187836899467480544568, −2.46662153128984277658460740372, −0.876310951112494298634045757971, 1.25061755204699369555888629357, 2.40524743672256137011116841790, 2.77091021599380360582694792293, 3.80312083762382190838890465070, 4.96603081721109267159462794583, 5.20032903992918423734641365957, 6.22698090917270415767630985565, 7.37858735152002197810473691779, 8.119482517484595697113128409814, 8.685014678482404911651142867152

Graph of the ZZ-function along the critical line