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Properties

Label 2-368-16.13-c1-0-13
Degree 22
Conductor 368368
Sign 0.0777+0.996i0.0777 + 0.996i
Analytic cond. 2.938492.93849
Root an. cond. 1.714201.71420
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  + (−1.40 − 0.111i)2-s + (−1.70 − 1.70i)3-s + (1.97 + 0.313i)4-s + (−2.61 + 2.61i)5-s + (2.21 + 2.59i)6-s + 1.66i·7-s + (−2.74 − 0.661i)8-s + 2.81i·9-s + (3.97 − 3.39i)10-s + (3.35 − 3.35i)11-s + (−2.83 − 3.90i)12-s + (1.50 + 1.50i)13-s + (0.184 − 2.34i)14-s + 8.91·15-s + (3.80 + 1.23i)16-s − 0.812·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0786i)2-s + (−0.984 − 0.984i)3-s + (0.987 + 0.156i)4-s + (−1.16 + 1.16i)5-s + (0.904 + 1.05i)6-s + 0.628i·7-s + (−0.972 − 0.233i)8-s + 0.939i·9-s + (1.25 − 1.07i)10-s + (1.01 − 1.01i)11-s + (−0.818 − 1.12i)12-s + (0.418 + 0.418i)13-s + (0.0493 − 0.626i)14-s + 2.30·15-s + (0.950 + 0.309i)16-s − 0.196·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((0.0777+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0777 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((0.0777+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0777 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: 0.0777+0.996i0.0777 + 0.996i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(93,⋅)\chi_{368} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), 0.0777+0.996i)(2,\ 368,\ (\ :1/2),\ 0.0777 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.294467−0.272392i0.294467 - 0.272392i
L(12)L(\frac12) ≈\approx 0.294467−0.272392i0.294467 - 0.272392i
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+(1.40+0.111i)T 1 + (1.40 + 0.111i)T
23 1+iT 1 + iT
good3 1+(1.70+1.70i)T+3iT2 1 + (1.70 + 1.70i)T + 3iT^{2}
5 1+(2.61−2.61i)T−5iT2 1 + (2.61 - 2.61i)T - 5iT^{2}
7 1−1.66iT−7T2 1 - 1.66iT - 7T^{2}
11 1+(−3.35+3.35i)T−11iT2 1 + (-3.35 + 3.35i)T - 11iT^{2}
13 1+(−1.50−1.50i)T+13iT2 1 + (-1.50 - 1.50i)T + 13iT^{2}
17 1+0.812T+17T2 1 + 0.812T + 17T^{2}
19 1+(3.81+3.81i)T+19iT2 1 + (3.81 + 3.81i)T + 19iT^{2}
29 1+(7.12+7.12i)T+29iT2 1 + (7.12 + 7.12i)T + 29iT^{2}
31 1−10.7T+31T2 1 - 10.7T + 31T^{2}
37 1+(−3.80+3.80i)T−37iT2 1 + (-3.80 + 3.80i)T - 37iT^{2}
41 1+0.765iT−41T2 1 + 0.765iT - 41T^{2}
43 1+(−3.75+3.75i)T−43iT2 1 + (-3.75 + 3.75i)T - 43iT^{2}
47 1−2.18T+47T2 1 - 2.18T + 47T^{2}
53 1+(1.48−1.48i)T−53iT2 1 + (1.48 - 1.48i)T - 53iT^{2}
59 1+(−9.46+9.46i)T−59iT2 1 + (-9.46 + 9.46i)T - 59iT^{2}
61 1+(−0.442−0.442i)T+61iT2 1 + (-0.442 - 0.442i)T + 61iT^{2}
67 1+(7.97+7.97i)T+67iT2 1 + (7.97 + 7.97i)T + 67iT^{2}
71 1+2.88iT−71T2 1 + 2.88iT - 71T^{2}
73 1+7.28iT−73T2 1 + 7.28iT - 73T^{2}
79 1−1.36T+79T2 1 - 1.36T + 79T^{2}
83 1+(−6.09−6.09i)T+83iT2 1 + (-6.09 - 6.09i)T + 83iT^{2}
89 1−4.98iT−89T2 1 - 4.98iT - 89T^{2}
97 1+7.46T+97T2 1 + 7.46T + 97T^{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

−11.36023339320395373039222929496, −10.71211308878071921448422920217, −9.198611047181941922084593091216, −8.260359593971973369233927935158, −7.34442536780783345119192970329, −6.45553625620473697462845150933, −6.12398739869297156697227726859, −3.83606431292845019629782321294, −2.40091883385074974794375894198, −0.51860020902536311059119138090, 1.12437948643308533121759092715, 3.88829739888502651118569408199, 4.56243470660021899235010289080, 5.83364991249140065131747039028, 7.04586739252778877340792431526, 8.063351235401827858795710567235, 8.933457649980123186614866089630, 9.873064775063412052567195751739, 10.61597914844362072463904116888, 11.51932899380008917920709361566

Graph of the ZZ-function along the critical line