Properties

Label 2-21e2-63.25-c1-0-6
Degree 22
Conductor 441441
Sign 0.1540.987i-0.154 - 0.987i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  − 1.69·2-s + (0.175 + 1.72i)3-s + 0.888·4-s + (0.474 − 0.822i)5-s + (−0.298 − 2.92i)6-s + 1.88·8-s + (−2.93 + 0.605i)9-s + (−0.806 + 1.39i)10-s + (0.294 + 0.509i)11-s + (0.156 + 1.53i)12-s + (2.50 + 4.34i)13-s + (1.5 + 0.673i)15-s − 4.98·16-s + (3.79 − 6.56i)17-s + (4.99 − 1.02i)18-s + (2.23 + 3.86i)19-s + ⋯
L(s)  = 1  − 1.20·2-s + (0.101 + 0.994i)3-s + 0.444·4-s + (0.212 − 0.367i)5-s + (−0.121 − 1.19i)6-s + 0.667·8-s + (−0.979 + 0.201i)9-s + (−0.255 + 0.441i)10-s + (0.0886 + 0.153i)11-s + (0.0451 + 0.442i)12-s + (0.696 + 1.20i)13-s + (0.387 + 0.173i)15-s − 1.24·16-s + (0.919 − 1.59i)17-s + (1.17 − 0.242i)18-s + (0.511 + 0.886i)19-s + ⋯

Functional equation

Λ(s)=(441s/2ΓC(s)L(s)=((0.1540.987i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2ΓC(s+1/2)L(s)=((0.1540.987i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32723^{2} \cdot 7^{2}
Sign: 0.1540.987i-0.154 - 0.987i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(214,)\chi_{441} (214, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.1540.987i)(2,\ 441,\ (\ :1/2),\ -0.154 - 0.987i)

Particular Values

L(1)L(1) \approx 0.458210+0.535682i0.458210 + 0.535682i
L(12)L(\frac12) \approx 0.458210+0.535682i0.458210 + 0.535682i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.1751.72i)T 1 + (-0.175 - 1.72i)T
7 1 1
good2 1+1.69T+2T2 1 + 1.69T + 2T^{2}
5 1+(0.474+0.822i)T+(2.54.33i)T2 1 + (-0.474 + 0.822i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.2940.509i)T+(5.5+9.52i)T2 1 + (-0.294 - 0.509i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.504.34i)T+(6.5+11.2i)T2 1 + (-2.50 - 4.34i)T + (-6.5 + 11.2i)T^{2}
17 1+(3.79+6.56i)T+(8.514.7i)T2 1 + (-3.79 + 6.56i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.233.86i)T+(9.5+16.4i)T2 1 + (-2.23 - 3.86i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.232.14i)T+(11.519.9i)T2 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.734.74i)T+(14.525.1i)T2 1 + (2.73 - 4.74i)T + (-14.5 - 25.1i)T^{2}
31 1+6.07T+31T2 1 + 6.07T + 31T^{2}
37 1+(3.496.05i)T+(18.5+32.0i)T2 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.5270.913i)T+(20.5+35.5i)T2 1 + (-0.527 - 0.913i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.496.05i)T+(21.537.2i)T2 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2}
47 1+7.47T+47T2 1 + 7.47T + 47T^{2}
53 1+(3.465.99i)T+(26.545.8i)T2 1 + (3.46 - 5.99i)T + (-26.5 - 45.8i)T^{2}
59 110.4T+59T2 1 - 10.4T + 59T^{2}
61 111.6T+61T2 1 - 11.6T + 61T^{2}
67 1+11.8T+67T2 1 + 11.8T + 67T^{2}
71 14.30T+71T2 1 - 4.30T + 71T^{2}
73 1+(2.233.86i)T+(36.563.2i)T2 1 + (2.23 - 3.86i)T + (-36.5 - 63.2i)T^{2}
79 1+1.33T+79T2 1 + 1.33T + 79T^{2}
83 1+(2.84+4.92i)T+(41.571.8i)T2 1 + (-2.84 + 4.92i)T + (-41.5 - 71.8i)T^{2}
89 1+(0.421+0.730i)T+(44.5+77.0i)T2 1 + (0.421 + 0.730i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.70+2.94i)T+(48.584.0i)T2 1 + (-1.70 + 2.94i)T + (-48.5 - 84.0i)T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−11.22116197663704349037386080188, −10.05128031632673280274990252485, −9.499756555602935691705799805779, −8.998856828782080792625965019814, −8.037284899142455764759125072043, −7.04598163848948017495439506938, −5.54450183014960597903742450518, −4.61819611761172619370292157909, −3.37034421877396565540693327987, −1.48517922615807702866077263518, 0.72318971294014726046419862601, 2.09330994365511835033269380961, 3.57571208621274936039268884031, 5.51110293983403985803222925369, 6.44118221603083359049984824188, 7.50169149488329629351164359801, 8.182712498916013788561657975683, 8.817253749687301389657331984886, 9.985450854574256241010860144722, 10.70049880153713799999447312897

Graph of the ZZ-function along the critical line