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Properties

Label 2-637-91.74-c1-0-41
Degree 22
Conductor 637637
Sign −0.799−0.600i-0.799 - 0.600i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.707 − 1.22i)3-s − 4-s + (−2.04 − 3.54i)5-s + (−0.707 − 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (−2.04 − 3.54i)10-s + (1.89 + 3.28i)11-s + (0.707 + 1.22i)12-s + (−0.634 + 3.54i)13-s + (−2.89 + 5.01i)15-s − 16-s − 1.26·17-s + (0.500 − 0.866i)18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.707i)3-s − 0.5·4-s + (−0.916 − 1.58i)5-s + (−0.288 − 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.647 − 1.12i)10-s + (0.572 + 0.991i)11-s + (0.204 + 0.353i)12-s + (−0.176 + 0.984i)13-s + (−0.748 + 1.29i)15-s − 0.250·16-s − 0.307·17-s + (0.117 − 0.204i)18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.799−0.600i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.799−0.600i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.799−0.600i-0.799 - 0.600i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(165,⋅)\chi_{637} (165, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.799−0.600i)(2,\ 637,\ (\ :1/2),\ -0.799 - 0.600i)

Particular Values

L(1)L(1) ≈\approx 0.127386+0.381454i0.127386 + 0.381454i
L(12)L(\frac12) ≈\approx 0.127386+0.381454i0.127386 + 0.381454i
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+(0.634−3.54i)T 1 + (0.634 - 3.54i)T
good2 1−T+2T2 1 - T + 2T^{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+(2.04+3.54i)T+(−2.5+4.33i)T2 1 + (2.04 + 3.54i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.89−3.28i)T+(−5.5+9.52i)T2 1 + (-1.89 - 3.28i)T + (-5.5 + 9.52i)T^{2}
17 1+1.26T+17T2 1 + 1.26T + 17T^{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+7.79T+23T2 1 + 7.79T + 23T^{2}
29 1+(0.397−0.689i)T+(−14.5−25.1i)T2 1 + (0.397 - 0.689i)T + (-14.5 - 25.1i)T^{2}
31 1+(−0.707+1.22i)T+(−15.5−26.8i)T2 1 + (-0.707 + 1.22i)T + (-15.5 - 26.8i)T^{2}
37 1−2.79T+37T2 1 - 2.79T + 37T^{2}
41 1+(1.48−2.57i)T+(−20.5−35.5i)T2 1 + (1.48 - 2.57i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.89+6.75i)T+(−21.5+37.2i)T2 1 + (3.89 + 6.75i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.41+2.44i)T+(−23.5+40.7i)T2 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6.29+10.9i)T+(−26.5−45.8i)T2 1 + (-6.29 + 10.9i)T + (-26.5 - 45.8i)T^{2}
59 1+12.4T+59T2 1 + 12.4T + 59T^{2}
61 1+(4.17−7.22i)T+(−30.5−52.8i)T2 1 + (4.17 - 7.22i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.89−3.28i)T+(−33.5+58.0i)T2 1 + (-1.89 - 3.28i)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+(−6.29+10.8i)T+(−36.5−63.2i)T2 1 + (-6.29 + 10.8i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.10−1.90i)T+(−39.5+68.4i)T2 1 + (-1.10 - 1.90i)T + (-39.5 + 68.4i)T^{2}
83 1+9.89T+83T2 1 + 9.89T + 83T^{2}
89 1+14.9T+89T2 1 + 14.9T + 89T^{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}
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   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

−9.727996011780066538386788410682, −9.177594647178136452247083294109, −8.329712509896529966060692955423, −7.31806460269197796324218810692, −6.36158300534406703796087022537, −5.23057603277578183355679656986, −4.37222800816157669300173479210, −3.90354171036263999738015430229, −1.65642452258053526831642049145, −0.18803524849220343707046100301, 2.91023719803409712959394488642, 3.70944062101175525304037601134, 4.39011797812131996683950273538, 5.68200498818917686389429261663, 6.30984099576389531427242925532, 7.62287393438997108172910761964, 8.311582504875375206923165283830, 9.702433291958271798618058320643, 10.36898164826039010998662897138, 11.14395355723030652978511348550

Graph of the ZZ-function along the critical line