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Properties

Label 2-384-128.109-c1-0-14
Degree 22
Conductor 384384
Sign −0.338−0.940i-0.338 - 0.940i
Analytic cond. 3.066253.06625
Root an. cond. 1.751071.75107
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  + (0.986 + 1.01i)2-s + (0.881 + 0.471i)3-s + (−0.0550 + 1.99i)4-s + (−0.677 + 0.825i)5-s + (0.391 + 1.35i)6-s + (1.41 + 0.942i)7-s + (−2.08 + 1.91i)8-s + (0.555 + 0.831i)9-s + (−1.50 + 0.127i)10-s + (−1.74 − 0.528i)11-s + (−0.991 + 1.73i)12-s + (2.10 − 1.72i)13-s + (0.435 + 2.35i)14-s + (−0.987 + 0.408i)15-s + (−3.99 − 0.220i)16-s + (−1.28 − 0.531i)17-s + ⋯
L(s)  = 1  + (0.697 + 0.716i)2-s + (0.509 + 0.272i)3-s + (−0.0275 + 0.999i)4-s + (−0.303 + 0.369i)5-s + (0.159 + 0.554i)6-s + (0.532 + 0.356i)7-s + (−0.735 + 0.677i)8-s + (0.185 + 0.277i)9-s + (−0.476 + 0.0402i)10-s + (−0.525 − 0.159i)11-s + (−0.286 + 0.501i)12-s + (0.582 − 0.478i)13-s + (0.116 + 0.630i)14-s + (−0.254 + 0.105i)15-s + (−0.998 − 0.0550i)16-s + (−0.311 − 0.128i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((−0.338−0.940i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((−0.338−0.940i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: −0.338−0.940i-0.338 - 0.940i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(109,⋅)\chi_{384} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), −0.338−0.940i)(2,\ 384,\ (\ :1/2),\ -0.338 - 0.940i)

Particular Values

L(1)L(1) ≈\approx 1.24500+1.77110i1.24500 + 1.77110i
L(12)L(\frac12) ≈\approx 1.24500+1.77110i1.24500 + 1.77110i
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+(−0.986−1.01i)T 1 + (-0.986 - 1.01i)T
3 1+(−0.881−0.471i)T 1 + (-0.881 - 0.471i)T
good5 1+(0.677−0.825i)T+(−0.975−4.90i)T2 1 + (0.677 - 0.825i)T + (-0.975 - 4.90i)T^{2}
7 1+(−1.41−0.942i)T+(2.67+6.46i)T2 1 + (-1.41 - 0.942i)T + (2.67 + 6.46i)T^{2}
11 1+(1.74+0.528i)T+(9.14+6.11i)T2 1 + (1.74 + 0.528i)T + (9.14 + 6.11i)T^{2}
13 1+(−2.10+1.72i)T+(2.53−12.7i)T2 1 + (-2.10 + 1.72i)T + (2.53 - 12.7i)T^{2}
17 1+(1.28+0.531i)T+(12.0+12.0i)T2 1 + (1.28 + 0.531i)T + (12.0 + 12.0i)T^{2}
19 1+(−0.136+1.38i)T+(−18.6−3.70i)T2 1 + (-0.136 + 1.38i)T + (-18.6 - 3.70i)T^{2}
23 1+(−3.76−0.749i)T+(21.2+8.80i)T2 1 + (-3.76 - 0.749i)T + (21.2 + 8.80i)T^{2}
29 1+(1.26+4.16i)T+(−24.1+16.1i)T2 1 + (1.26 + 4.16i)T + (-24.1 + 16.1i)T^{2}
31 1+(−0.429+0.429i)T−31iT2 1 + (-0.429 + 0.429i)T - 31iT^{2}
37 1+(−9.15+0.902i)T+(36.2−7.21i)T2 1 + (-9.15 + 0.902i)T + (36.2 - 7.21i)T^{2}
41 1+(−0.0409+0.205i)T+(−37.8−15.6i)T2 1 + (-0.0409 + 0.205i)T + (-37.8 - 15.6i)T^{2}
43 1+(−0.656+0.350i)T+(23.8−35.7i)T2 1 + (-0.656 + 0.350i)T + (23.8 - 35.7i)T^{2}
47 1+(1.66−4.01i)T+(−33.2−33.2i)T2 1 + (1.66 - 4.01i)T + (-33.2 - 33.2i)T^{2}
53 1+(−2.99+9.86i)T+(−44.0−29.4i)T2 1 + (-2.99 + 9.86i)T + (-44.0 - 29.4i)T^{2}
59 1+(1.91+1.57i)T+(11.5+57.8i)T2 1 + (1.91 + 1.57i)T + (11.5 + 57.8i)T^{2}
61 1+(−6.69+12.5i)T+(−33.8−50.7i)T2 1 + (-6.69 + 12.5i)T + (-33.8 - 50.7i)T^{2}
67 1+(5.87−10.9i)T+(−37.2−55.7i)T2 1 + (5.87 - 10.9i)T + (-37.2 - 55.7i)T^{2}
71 1+(1.22−1.83i)T+(−27.1−65.5i)T2 1 + (1.22 - 1.83i)T + (-27.1 - 65.5i)T^{2}
73 1+(−4.58+3.06i)T+(27.9−67.4i)T2 1 + (-4.58 + 3.06i)T + (27.9 - 67.4i)T^{2}
79 1+(−2.27−5.49i)T+(−55.8+55.8i)T2 1 + (-2.27 - 5.49i)T + (-55.8 + 55.8i)T^{2}
83 1+(7.02+0.691i)T+(81.4+16.1i)T2 1 + (7.02 + 0.691i)T + (81.4 + 16.1i)T^{2}
89 1+(14.5−2.89i)T+(82.2−34.0i)T2 1 + (14.5 - 2.89i)T + (82.2 - 34.0i)T^{2}
97 1+(−7.28+7.28i)T−97iT2 1 + (-7.28 + 7.28i)T - 97iT^{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

−11.48957226474368456594356960873, −11.04476330351685157171844998532, −9.585608222422790701792064957093, −8.526341491186007356861932140910, −7.896576152031804732350559159583, −6.95242469739416326902466659184, −5.72774436141476622855915888569, −4.79626196138047377357566560957, −3.61900774918983177860402969574, −2.58031704336840290658352333519, 1.28450904500149154822904803008, 2.69073783774904859692863747365, 4.01908839830142223610661758730, 4.82020850013376936145445942416, 6.12279593196783262957449467140, 7.28596403164716075860441585977, 8.405604168614777756615093075802, 9.255350084311444488115264290387, 10.40786226298361561353566098163, 11.13850017092324574044274160445

Graph of the ZZ-function along the critical line