↑

Properties

Label 2-475-19.7-c1-0-14
Degree 22
Conductor 475475
Sign 0.0977−0.995i0.0977 - 0.995i
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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 + 1.73i)2-s + (−0.999 + 1.73i)4-s + 4·7-s + (1.5 − 2.59i)9-s − 11-s + (−1 + 1.73i)13-s + (4 + 6.92i)14-s + (1.99 + 3.46i)16-s + (−1 − 1.73i)17-s + 6·18-s + (−3.5 + 2.59i)19-s + (−1 − 1.73i)22-s + (3 − 5.19i)23-s − 3.99·26-s + (−3.99 + 6.92i)28-s + (−4.5 + 7.79i)29-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + 1.51·7-s + (0.5 − 0.866i)9-s − 0.301·11-s + (−0.277 + 0.480i)13-s + (1.06 + 1.85i)14-s + (0.499 + 0.866i)16-s + (−0.242 − 0.420i)17-s + 1.41·18-s + (−0.802 + 0.596i)19-s + (−0.213 − 0.369i)22-s + (0.625 − 1.08i)23-s − 0.784·26-s + (−0.755 + 1.30i)28-s + (−0.835 + 1.44i)29-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.0977−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.0977−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.0977−0.995i0.0977 - 0.995i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(26,⋅)\chi_{475} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), 0.0977−0.995i)(2,\ 475,\ (\ :1/2),\ 0.0977 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.75517+1.59125i1.75517 + 1.59125i
L(12)L(\frac12) ≈\approx 1.75517+1.59125i1.75517 + 1.59125i
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)
bad5 1 1
19 1+(3.5−2.59i)T 1 + (3.5 - 2.59i)T
good2 1+(−1−1.73i)T+(−1+1.73i)T2 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2}
3 1+(−1.5+2.59i)T2 1 + (-1.5 + 2.59i)T^{2}
7 1−4T+7T2 1 - 4T + 7T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1+(1−1.73i)T+(−6.5−11.2i)T2 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2}
17 1+(1+1.73i)T+(−8.5+14.7i)T2 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}
23 1+(−3+5.19i)T+(−11.5−19.9i)T2 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.5−7.79i)T+(−14.5−25.1i)T2 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2}
31 1+7T+31T2 1 + 7T + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+(1+1.73i)T+(−20.5+35.5i)T2 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2}
43 1+(−1−1.73i)T+(−21.5+37.2i)T2 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3+5.19i)T+(−23.5−40.7i)T2 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}
53 1+(−2+3.46i)T+(−26.5−45.8i)T2 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.5+7.79i)T+(−29.5+51.0i)T2 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.5+6.06i)T+(−30.5−52.8i)T2 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(5−8.66i)T+(−33.5−58.0i)T2 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2}
71 1+(0.5+0.866i)T+(−35.5+61.4i)T2 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2}
73 1+(−5−8.66i)T+(−36.5+63.2i)T2 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2}
79 1+(0.5+0.866i)T+(−39.5+68.4i)T2 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1+(−5.5+9.52i)T+(−44.5−77.0i)T2 1 + (-5.5 + 9.52i)T + (-44.5 - 77.0i)T^{2}
97 1+(3+5.19i)T+(−48.5+84.0i)T2 1 + (3 + 5.19i)T + (-48.5 + 84.0i)T^{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

−11.21206814392667914991093789148, −10.47926428389554250498594186373, −9.066647111327061047407457202854, −8.284883554916736610418978676258, −7.28682255474782080712788413973, −6.70759651690475235308707442941, −5.47336379223912505359138887712, −4.75128138598538874450553548668, −3.85089679886022452512250861402, −1.77629153868512302240926579949, 1.60205579459502770893204042843, 2.45324272264237617901690965065, 3.98980113475805215523919362745, 4.80195433734292026316143436394, 5.53545734999154402714993023198, 7.43139367874779369906396395452, 7.935791464766070634966570650783, 9.231403414712642924893437783538, 10.47932195585320547667850945242, 10.90471578686655770881722212476

Graph of the ZZ-function along the critical line