↑

Properties

Label 2-637-7.2-c1-0-34
Degree 22
Conductor 637637
Sign −0.900+0.435i-0.900 + 0.435i
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  + (1.22 − 2.11i)2-s + (0.333 + 0.578i)3-s + (−1.99 − 3.45i)4-s + (−0.455 + 0.788i)5-s + 1.63·6-s − 4.87·8-s + (1.27 − 2.21i)9-s + (1.11 + 1.92i)10-s + (−1.83 − 3.18i)11-s + (1.33 − 2.30i)12-s + 13-s − 0.607·15-s + (−1.97 + 3.42i)16-s + (−3.59 − 6.22i)17-s + (−3.12 − 5.41i)18-s + (0.989 − 1.71i)19-s + ⋯
L(s)  = 1  + (0.865 − 1.49i)2-s + (0.192 + 0.333i)3-s + (−0.997 − 1.72i)4-s + (−0.203 + 0.352i)5-s + 0.667·6-s − 1.72·8-s + (0.425 − 0.737i)9-s + (0.352 + 0.610i)10-s + (−0.554 − 0.960i)11-s + (0.384 − 0.666i)12-s + 0.277·13-s − 0.156·15-s + (−0.493 + 0.855i)16-s + (−0.871 − 1.50i)17-s + (−0.736 − 1.27i)18-s + (0.226 − 0.392i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.900+0.435i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.900+0.435i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.900+0.435i-0.900 + 0.435i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(79,⋅)\chi_{637} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.900+0.435i)(2,\ 637,\ (\ :1/2),\ -0.900 + 0.435i)

Particular Values

L(1)L(1) ≈\approx 0.478176−2.08458i0.478176 - 2.08458i
L(12)L(\frac12) ≈\approx 0.478176−2.08458i0.478176 - 2.08458i
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−T 1 - T
good2 1+(−1.22+2.11i)T+(−1−1.73i)T2 1 + (-1.22 + 2.11i)T + (-1 - 1.73i)T^{2}
3 1+(−0.333−0.578i)T+(−1.5+2.59i)T2 1 + (-0.333 - 0.578i)T + (-1.5 + 2.59i)T^{2}
5 1+(0.455−0.788i)T+(−2.5−4.33i)T2 1 + (0.455 - 0.788i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.83+3.18i)T+(−5.5+9.52i)T2 1 + (1.83 + 3.18i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.59+6.22i)T+(−8.5+14.7i)T2 1 + (3.59 + 6.22i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.989+1.71i)T+(−9.5−16.4i)T2 1 + (-0.989 + 1.71i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.298+0.516i)T+(−11.5−19.9i)T2 1 + (-0.298 + 0.516i)T + (-11.5 - 19.9i)T^{2}
29 1+3.64T+29T2 1 + 3.64T + 29T^{2}
31 1+(−3.54−6.13i)T+(−15.5+26.8i)T2 1 + (-3.54 - 6.13i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.355−0.615i)T+(−18.5−32.0i)T2 1 + (0.355 - 0.615i)T + (-18.5 - 32.0i)T^{2}
41 1−5.27T+41T2 1 - 5.27T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1+(6.05−10.4i)T+(−23.5−40.7i)T2 1 + (6.05 - 10.4i)T + (-23.5 - 40.7i)T^{2}
53 1+(−5.72−9.91i)T+(−26.5+45.8i)T2 1 + (-5.72 - 9.91i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.79+8.30i)T+(−29.5+51.0i)T2 1 + (4.79 + 8.30i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.49−6.04i)T+(−30.5−52.8i)T2 1 + (3.49 - 6.04i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.614+1.06i)T+(−33.5+58.0i)T2 1 + (0.614 + 1.06i)T + (-33.5 + 58.0i)T^{2}
71 1−11.3T+71T2 1 - 11.3T + 71T^{2}
73 1+(3.26+5.65i)T+(−36.5+63.2i)T2 1 + (3.26 + 5.65i)T + (-36.5 + 63.2i)T^{2}
79 1+(−5.76+9.97i)T+(−39.5−68.4i)T2 1 + (-5.76 + 9.97i)T + (-39.5 - 68.4i)T^{2}
83 1−7.16T+83T2 1 - 7.16T + 83T^{2}
89 1+(6.42−11.1i)T+(−44.5−77.0i)T2 1 + (6.42 - 11.1i)T + (-44.5 - 77.0i)T^{2}
97 1−9.09T+97T2 1 - 9.09T + 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

−10.73063239336933365071981635093, −9.417606197898604867397568191891, −9.086838859231880776956249748600, −7.53111357967609151535754372159, −6.35411399190014041675806641152, −5.17547438041021622959405744268, −4.33329316621370691391763063256, −3.27526661618185708618707421708, −2.69861898932120978669218945956, −0.911221143111126059948230595038, 2.13477671766083700704479983465, 3.95796081167060601758589634014, 4.59098328407437423023924570829, 5.57364902169326114957576441868, 6.50359254536394360696330630464, 7.37885473308904339858733085077, 8.026084917981521573843772301304, 8.655197350739499023368567903055, 9.982016527578675106929996765347, 10.97190005761895829541595242678

Graph of the ZZ-function along the critical line