↑

Properties

Label 2-637-13.3-c1-0-35
Degree 22
Conductor 637637
Sign 0.0910+0.995i0.0910 + 0.995i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
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  + (−0.5 − 0.866i)2-s + (0.707 + 1.22i)3-s + (0.500 − 0.866i)4-s + 2.68·5-s + (0.707 − 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (−1.34 − 2.32i)10-s + (−2.89 − 5.01i)11-s + 1.41·12-s + (−2.75 − 2.32i)13-s + (1.89 + 3.28i)15-s + (0.500 + 0.866i)16-s + (2.75 − 4.77i)17-s − 1.00·18-s + (−1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.408 + 0.707i)3-s + (0.250 − 0.433i)4-s + 1.20·5-s + (0.288 − 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.424 − 0.735i)10-s + (−0.873 − 1.51i)11-s + 0.408·12-s + (−0.764 − 0.644i)13-s + (0.490 + 0.848i)15-s + (0.125 + 0.216i)16-s + (0.668 − 1.15i)17-s − 0.235·18-s + (−0.324 + 0.561i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.0910+0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.0910+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.0910+0.995i0.0910 + 0.995i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(393,⋅)\chi_{637} (393, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.0910+0.995i)(2,\ 637,\ (\ :1/2),\ 0.0910 + 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.23445−1.12671i1.23445 - 1.12671i
L(12)L(\frac12) ≈\approx 1.23445−1.12671i1.23445 - 1.12671i
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)
bad7 1 1
13 1+(2.75+2.32i)T 1 + (2.75 + 2.32i)T
good2 1+(0.5+0.866i)T+(−1+1.73i)T2 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2}
3 1+(−0.707−1.22i)T+(−1.5+2.59i)T2 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2}
5 1−2.68T+5T2 1 - 2.68T + 5T^{2}
11 1+(2.89+5.01i)T+(−5.5+9.52i)T2 1 + (2.89 + 5.01i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.75+4.77i)T+(−8.5−14.7i)T2 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.41−2.44i)T+(−9.5−16.4i)T2 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.897+1.55i)T+(−11.5+19.9i)T2 1 + (0.897 + 1.55i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.39−7.61i)T+(−14.5+25.1i)T2 1 + (-4.39 - 7.61i)T + (-14.5 + 25.1i)T^{2}
31 1−1.41T+31T2 1 - 1.41T + 31T^{2}
37 1+(−3.39−5.88i)T+(−18.5+32.0i)T2 1 + (-3.39 - 5.88i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4.87−8.44i)T+(−20.5+35.5i)T2 1 + (-4.87 - 8.44i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.897+1.55i)T+(−21.5−37.2i)T2 1 + (-0.897 + 1.55i)T + (-21.5 - 37.2i)T^{2}
47 1+2.82T+47T2 1 + 2.82T + 47T^{2}
53 1−6.59T+53T2 1 - 6.59T + 53T^{2}
59 1+(−0.562+0.974i)T+(−29.5−51.0i)T2 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.779+1.34i)T+(−30.5−52.8i)T2 1 + (-0.779 + 1.34i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.89+5.01i)T+(−33.5+58.0i)T2 1 + (2.89 + 5.01i)T + (-33.5 + 58.0i)T^{2}
71 1+(3−5.19i)T+(−35.5−61.4i)T2 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1−5.80T+73T2 1 - 5.80T + 73T^{2}
79 1+11.7T+79T2 1 + 11.7T + 79T^{2}
83 1−9.89T+83T2 1 - 9.89T + 83T^{2}
89 1+(−6.07−10.5i)T+(−44.5+77.0i)T2 1 + (-6.07 - 10.5i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.12−3.67i)T+(−48.5−84.0i)T2 1 + (2.12 - 3.67i)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

−10.19412828352523311799795104499, −9.786578984645920463775055461304, −9.008823285113054234305765720158, −8.080714682681255716862561081337, −6.59081439986157425309242539116, −5.72702079488706144536748750665, −4.99096804346814501108726176686, −3.15993074784555366260208913196, −2.65024334726365737682228740939, −0.977760354114844897107363975100, 2.05312526508072952704541987436, 2.46150903536508792741100307748, 4.38238361967451160998811537737, 5.63698870430492505635840110897, 6.55759822120836391596107954792, 7.40859353588782133375642776116, 7.85669017385208443750199452557, 8.930681543370470385883079646928, 9.835032416503586328604294107453, 10.41882806740735312595596705281

Graph of the ZZ-function along the critical line