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Properties

Label 2-384-128.101-c1-0-15
Degree 22
Conductor 384384
Sign 0.446−0.894i0.446 - 0.894i
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  + (1.19 + 0.755i)2-s + (0.881 − 0.471i)3-s + (0.857 + 1.80i)4-s + (0.324 + 0.396i)5-s + (1.41 + 0.103i)6-s + (−2.62 + 1.75i)7-s + (−0.341 + 2.80i)8-s + (0.555 − 0.831i)9-s + (0.0890 + 0.718i)10-s + (2.66 − 0.809i)11-s + (1.60 + 1.18i)12-s + (3.50 + 2.87i)13-s + (−4.46 + 0.112i)14-s + (0.473 + 0.196i)15-s + (−2.53 + 3.09i)16-s + (−1.24 + 0.514i)17-s + ⋯
L(s)  = 1  + (0.845 + 0.534i)2-s + (0.509 − 0.272i)3-s + (0.428 + 0.903i)4-s + (0.145 + 0.177i)5-s + (0.575 + 0.0421i)6-s + (−0.992 + 0.662i)7-s + (−0.120 + 0.992i)8-s + (0.185 − 0.277i)9-s + (0.0281 + 0.227i)10-s + (0.804 − 0.243i)11-s + (0.464 + 0.343i)12-s + (0.973 + 0.798i)13-s + (−1.19 + 0.0299i)14-s + (0.122 + 0.0506i)15-s + (−0.632 + 0.774i)16-s + (−0.301 + 0.124i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.446−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.446−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.446−0.894i0.446 - 0.894i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(229,⋅)\chi_{384} (229, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), 0.446−0.894i)(2,\ 384,\ (\ :1/2),\ 0.446 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 2.08197+1.28836i2.08197 + 1.28836i
L(12)L(\frac12) ≈\approx 2.08197+1.28836i2.08197 + 1.28836i
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+(−1.19−0.755i)T 1 + (-1.19 - 0.755i)T
3 1+(−0.881+0.471i)T 1 + (-0.881 + 0.471i)T
good5 1+(−0.324−0.396i)T+(−0.975+4.90i)T2 1 + (-0.324 - 0.396i)T + (-0.975 + 4.90i)T^{2}
7 1+(2.62−1.75i)T+(2.67−6.46i)T2 1 + (2.62 - 1.75i)T + (2.67 - 6.46i)T^{2}
11 1+(−2.66+0.809i)T+(9.14−6.11i)T2 1 + (-2.66 + 0.809i)T + (9.14 - 6.11i)T^{2}
13 1+(−3.50−2.87i)T+(2.53+12.7i)T2 1 + (-3.50 - 2.87i)T + (2.53 + 12.7i)T^{2}
17 1+(1.24−0.514i)T+(12.0−12.0i)T2 1 + (1.24 - 0.514i)T + (12.0 - 12.0i)T^{2}
19 1+(0.508+5.16i)T+(−18.6+3.70i)T2 1 + (0.508 + 5.16i)T + (-18.6 + 3.70i)T^{2}
23 1+(−0.610+0.121i)T+(21.2−8.80i)T2 1 + (-0.610 + 0.121i)T + (21.2 - 8.80i)T^{2}
29 1+(−1.53+5.06i)T+(−24.1−16.1i)T2 1 + (-1.53 + 5.06i)T + (-24.1 - 16.1i)T^{2}
31 1+(4.71+4.71i)T+31iT2 1 + (4.71 + 4.71i)T + 31iT^{2}
37 1+(0.337+0.0332i)T+(36.2+7.21i)T2 1 + (0.337 + 0.0332i)T + (36.2 + 7.21i)T^{2}
41 1+(0.532+2.67i)T+(−37.8+15.6i)T2 1 + (0.532 + 2.67i)T + (-37.8 + 15.6i)T^{2}
43 1+(4.98+2.66i)T+(23.8+35.7i)T2 1 + (4.98 + 2.66i)T + (23.8 + 35.7i)T^{2}
47 1+(−2.42−5.85i)T+(−33.2+33.2i)T2 1 + (-2.42 - 5.85i)T + (-33.2 + 33.2i)T^{2}
53 1+(−3.19−10.5i)T+(−44.0+29.4i)T2 1 + (-3.19 - 10.5i)T + (-44.0 + 29.4i)T^{2}
59 1+(10.5−8.66i)T+(11.5−57.8i)T2 1 + (10.5 - 8.66i)T + (11.5 - 57.8i)T^{2}
61 1+(1.48+2.77i)T+(−33.8+50.7i)T2 1 + (1.48 + 2.77i)T + (-33.8 + 50.7i)T^{2}
67 1+(5.88+11.0i)T+(−37.2+55.7i)T2 1 + (5.88 + 11.0i)T + (-37.2 + 55.7i)T^{2}
71 1+(4.98+7.45i)T+(−27.1+65.5i)T2 1 + (4.98 + 7.45i)T + (-27.1 + 65.5i)T^{2}
73 1+(−6.36−4.25i)T+(27.9+67.4i)T2 1 + (-6.36 - 4.25i)T + (27.9 + 67.4i)T^{2}
79 1+(−3.81+9.21i)T+(−55.8−55.8i)T2 1 + (-3.81 + 9.21i)T + (-55.8 - 55.8i)T^{2}
83 1+(−12.4+1.22i)T+(81.4−16.1i)T2 1 + (-12.4 + 1.22i)T + (81.4 - 16.1i)T^{2}
89 1+(−12.3−2.45i)T+(82.2+34.0i)T2 1 + (-12.3 - 2.45i)T + (82.2 + 34.0i)T^{2}
97 1+(4.31+4.31i)T+97iT2 1 + (4.31 + 4.31i)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

−11.80726465088447801809203936952, −10.81925962432706882078873813872, −9.171042501514572017787818352499, −8.882570928759017211725390472648, −7.54956972659997407885009421772, −6.33921029106982572069739512610, −6.22523988090503505899846601603, −4.46362214801544550403374400658, −3.42269140370008155292988479275, −2.31641331963299723725309191666, 1.47746986677270387049355725629, 3.31728821969332947459250254220, 3.76382713583253674623070498112, 5.14657035866900396714108742925, 6.29294771571079693766603621632, 7.14874762159738996609566663762, 8.635464121696420418617630915017, 9.613535169650799520918292629332, 10.32530705249393316551952674440, 11.09986252947441547157263552603

Graph of the ZZ-function along the critical line