↑

Properties

Label 2-392-392.109-c1-0-20
Degree 22
Conductor 392392
Sign 0.0211−0.999i0.0211 - 0.999i
Analytic cond. 3.130133.13013
Root an. cond. 1.769211.76921
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.751 + 1.19i)2-s + (−0.115 − 0.768i)3-s + (−0.869 + 1.80i)4-s + (3.94 + 1.54i)5-s + (0.833 − 0.716i)6-s + (−2.02 + 1.70i)7-s + (−2.81 + 0.313i)8-s + (2.28 − 0.706i)9-s + (1.11 + 5.88i)10-s + (0.256 − 0.829i)11-s + (1.48 + 0.459i)12-s + (−3.90 + 0.892i)13-s + (−3.56 − 1.14i)14-s + (0.732 − 3.21i)15-s + (−2.48 − 3.13i)16-s + (4.54 − 3.09i)17-s + ⋯
L(s)  = 1  + (0.531 + 0.846i)2-s + (−0.0668 − 0.443i)3-s + (−0.434 + 0.900i)4-s + (1.76 + 0.692i)5-s + (0.340 − 0.292i)6-s + (−0.766 + 0.642i)7-s + (−0.993 + 0.110i)8-s + (0.763 − 0.235i)9-s + (0.351 + 1.86i)10-s + (0.0771 − 0.250i)11-s + (0.428 + 0.132i)12-s + (−1.08 + 0.247i)13-s + (−0.951 − 0.307i)14-s + (0.189 − 0.829i)15-s + (−0.622 − 0.782i)16-s + (1.10 − 0.750i)17-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=((0.0211−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0211 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+1/2) L(s)=((0.0211−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0211 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: 0.0211−0.999i0.0211 - 0.999i
Analytic conductor: 3.130133.13013
Root analytic conductor: 1.769211.76921
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ392(109,⋅)\chi_{392} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 392, ( :1/2), 0.0211−0.999i)(2,\ 392,\ (\ :1/2),\ 0.0211 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.44705+1.41679i1.44705 + 1.41679i
L(12)L(\frac12) ≈\approx 1.44705+1.41679i1.44705 + 1.41679i
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+(−0.751−1.19i)T 1 + (-0.751 - 1.19i)T
7 1+(2.02−1.70i)T 1 + (2.02 - 1.70i)T
good3 1+(0.115+0.768i)T+(−2.86+0.884i)T2 1 + (0.115 + 0.768i)T + (-2.86 + 0.884i)T^{2}
5 1+(−3.94−1.54i)T+(3.66+3.40i)T2 1 + (-3.94 - 1.54i)T + (3.66 + 3.40i)T^{2}
11 1+(−0.256+0.829i)T+(−9.08−6.19i)T2 1 + (-0.256 + 0.829i)T + (-9.08 - 6.19i)T^{2}
13 1+(3.90−0.892i)T+(11.7−5.64i)T2 1 + (3.90 - 0.892i)T + (11.7 - 5.64i)T^{2}
17 1+(−4.54+3.09i)T+(6.21−15.8i)T2 1 + (-4.54 + 3.09i)T + (6.21 - 15.8i)T^{2}
19 1+(2.86−1.65i)T+(9.5−16.4i)T2 1 + (2.86 - 1.65i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.63−1.79i)T+(8.40+21.4i)T2 1 + (-2.63 - 1.79i)T + (8.40 + 21.4i)T^{2}
29 1+(−0.706+1.46i)T+(−18.0−22.6i)T2 1 + (-0.706 + 1.46i)T + (-18.0 - 22.6i)T^{2}
31 1+(2.09−3.62i)T+(−15.5−26.8i)T2 1 + (2.09 - 3.62i)T + (-15.5 - 26.8i)T^{2}
37 1+(6.13−0.459i)T+(36.5−5.51i)T2 1 + (6.13 - 0.459i)T + (36.5 - 5.51i)T^{2}
41 1+(5.02+6.29i)T+(−9.12+39.9i)T2 1 + (5.02 + 6.29i)T + (-9.12 + 39.9i)T^{2}
43 1+(−2.78−2.22i)T+(9.56+41.9i)T2 1 + (-2.78 - 2.22i)T + (9.56 + 41.9i)T^{2}
47 1+(−9.02+8.37i)T+(3.51−46.8i)T2 1 + (-9.02 + 8.37i)T + (3.51 - 46.8i)T^{2}
53 1+(1.66+0.124i)T+(52.4+7.89i)T2 1 + (1.66 + 0.124i)T + (52.4 + 7.89i)T^{2}
59 1+(−11.8+4.63i)T+(43.2−40.1i)T2 1 + (-11.8 + 4.63i)T + (43.2 - 40.1i)T^{2}
61 1+(0.0423−0.00317i)T+(60.3−9.09i)T2 1 + (0.0423 - 0.00317i)T + (60.3 - 9.09i)T^{2}
67 1+(9.81+5.66i)T+(33.5+58.0i)T2 1 + (9.81 + 5.66i)T + (33.5 + 58.0i)T^{2}
71 1+(−6.37+3.06i)T+(44.2−55.5i)T2 1 + (-6.37 + 3.06i)T + (44.2 - 55.5i)T^{2}
73 1+(8.10+7.51i)T+(5.45+72.7i)T2 1 + (8.10 + 7.51i)T + (5.45 + 72.7i)T^{2}
79 1+(−3.55−6.14i)T+(−39.5+68.4i)T2 1 + (-3.55 - 6.14i)T + (-39.5 + 68.4i)T^{2}
83 1+(14.1+3.22i)T+(74.7+36.0i)T2 1 + (14.1 + 3.22i)T + (74.7 + 36.0i)T^{2}
89 1+(−0.0985+0.0303i)T+(73.5−50.1i)T2 1 + (-0.0985 + 0.0303i)T + (73.5 - 50.1i)T^{2}
97 1+2.98T+97T2 1 + 2.98T + 97T^{2}
show more
show less
   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

−12.00505309367989537863566448157, −10.28304124478458415275057724081, −9.646011781458590819487257498480, −8.875394000689423154980004184846, −7.23649258881117701089389133220, −6.79174047237643294568683037439, −5.85922907106951690064513912673, −5.17623115230079877022606887887, −3.33142999054634528370217334915, −2.20087643957166395769760504907, 1.34824951724844566143757913632, 2.64028174918018929882009166472, 4.17038488859670098488302106321, 5.07846965686699471739767310639, 5.92026059159427433611185720939, 7.03403790599937256617756589201, 8.869540990920937702821205164624, 9.763953876992225533287212107608, 10.09069978499822193269058404755, 10.69159849003910738661937070452

Graph of the ZZ-function along the critical line