↑

Properties

Label 2-637-91.81-c1-0-23
Degree 22
Conductor 637637
Sign −0.113−0.993i-0.113 - 0.993i
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 + 3·3-s + (0.500 + 0.866i)4-s + (1.5 + 2.59i)5-s + (−1.5 + 2.59i)6-s − 3·8-s + 6·9-s − 3·10-s − 3·11-s + (1.50 + 2.59i)12-s + (1 − 3.46i)13-s + (4.5 + 7.79i)15-s + (0.500 − 0.866i)16-s + (−1 − 1.73i)17-s + (−3 + 5.19i)18-s + 19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + 1.73·3-s + (0.250 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.612 + 1.06i)6-s − 1.06·8-s + 2·9-s − 0.948·10-s − 0.904·11-s + (0.433 + 0.749i)12-s + (0.277 − 0.960i)13-s + (1.16 + 2.01i)15-s + (0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + (−0.707 + 1.22i)18-s + 0.229·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.113−0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.113−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.113−0.993i-0.113 - 0.993i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(263,⋅)\chi_{637} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.113−0.993i)(2,\ 637,\ (\ :1/2),\ -0.113 - 0.993i)

Particular Values

L(1)L(1) ≈\approx 1.63638+1.83430i1.63638 + 1.83430i
L(12)L(\frac12) ≈\approx 1.63638+1.83430i1.63638 + 1.83430i
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+(−1+3.46i)T 1 + (-1 + 3.46i)T
good2 1+(0.5−0.866i)T+(−1−1.73i)T2 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2}
3 1−3T+3T2 1 - 3T + 3T^{2}
5 1+(−1.5−2.59i)T+(−2.5+4.33i)T2 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
17 1+(1+1.73i)T+(−8.5+14.7i)T2 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(3.5+6.06i)T+(−14.5+25.1i)T2 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.5+2.59i)T+(−15.5−26.8i)T2 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2}
37 1+(1−1.73i)T+(−18.5−32.0i)T2 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.5−2.59i)T+(−20.5+35.5i)T2 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.5+6.06i)T+(−21.5−37.2i)T2 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.5−0.866i)T+(−23.5+40.7i)T2 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.5−2.59i)T+(−26.5−45.8i)T2 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(2+3.46i)T+(−29.5+51.0i)T2 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2}
61 1−13T+61T2 1 - 13T + 61T^{2}
67 1+3T+67T2 1 + 3T + 67T^{2}
71 1+(6.5−11.2i)T+(−35.5−61.4i)T2 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2}
73 1+(6.5−11.2i)T+(−36.5−63.2i)T2 1 + (6.5 - 11.2i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.5−2.59i)T+(−39.5+68.4i)T2 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−3+5.19i)T+(−44.5−77.0i)T2 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1+(2.5−4.33i)T+(−48.5−84.0i)T2 1 + (2.5 - 4.33i)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

−10.42149659983095379738553630387, −9.778685184786831876448337038634, −8.900877084746194396536859604799, −8.007832841253923214119331855791, −7.56306674795424406332456218611, −6.72426177460275896711449148052, −5.65715707973529754724028725698, −3.80813654828250170717958005537, −2.79304304217424092196289102658, −2.43417156954458916176106786594, 1.45000882721024066151316439259, 2.16984235531105985857885849370, 3.28411039031949028525051511701, 4.60003071303646567570301607707, 5.71406097845946011556786003564, 7.00336509204686665321898464106, 8.139316254233997057346155697860, 8.989526376112955629874363703719, 9.225572483665266828648821705129, 10.08993635598612210373246806177

Graph of the ZZ-function along the critical line