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Properties

Label 2-637-7.2-c1-0-31
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.132 + 0.229i)2-s + (−1.45 − 2.51i)3-s + (0.964 + 1.67i)4-s + (0.717 − 1.24i)5-s + 0.769·6-s − 1.03·8-s + (−2.73 + 4.73i)9-s + (0.189 + 0.328i)10-s + (−2.75 − 4.76i)11-s + (2.80 − 4.86i)12-s + 13-s − 4.17·15-s + (−1.79 + 3.10i)16-s + (−2.41 − 4.18i)17-s + (−0.722 − 1.25i)18-s + (−1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + (−0.0935 + 0.162i)2-s + (−0.839 − 1.45i)3-s + (0.482 + 0.835i)4-s + (0.320 − 0.555i)5-s + 0.314·6-s − 0.367·8-s + (−0.910 + 1.57i)9-s + (0.0600 + 0.104i)10-s + (−0.829 − 1.43i)11-s + (0.810 − 1.40i)12-s + 0.277·13-s − 1.07·15-s + (−0.448 + 0.776i)16-s + (−0.585 − 1.01i)17-s + (−0.170 − 0.295i)18-s + (−0.323 + 0.560i)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 0.163431−0.712471i0.163431 - 0.712471i
L(12)L(\frac12) ≈\approx 0.163431−0.712471i0.163431 - 0.712471i
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+(0.132−0.229i)T+(−1−1.73i)T2 1 + (0.132 - 0.229i)T + (-1 - 1.73i)T^{2}
3 1+(1.45+2.51i)T+(−1.5+2.59i)T2 1 + (1.45 + 2.51i)T + (-1.5 + 2.59i)T^{2}
5 1+(−0.717+1.24i)T+(−2.5−4.33i)T2 1 + (-0.717 + 1.24i)T + (-2.5 - 4.33i)T^{2}
11 1+(2.75+4.76i)T+(−5.5+9.52i)T2 1 + (2.75 + 4.76i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.41+4.18i)T+(−8.5+14.7i)T2 1 + (2.41 + 4.18i)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+(−2.99+5.19i)T+(−11.5−19.9i)T2 1 + (-2.99 + 5.19i)T + (-11.5 - 19.9i)T^{2}
29 1−1.04T+29T2 1 - 1.04T + 29T^{2}
31 1+(4.60+7.97i)T+(−15.5+26.8i)T2 1 + (4.60 + 7.97i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.306−0.530i)T+(−18.5−32.0i)T2 1 + (0.306 - 0.530i)T + (-18.5 - 32.0i)T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1+8.43T+43T2 1 + 8.43T + 43T^{2}
47 1+(1.20−2.08i)T+(−23.5−40.7i)T2 1 + (1.20 - 2.08i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.914−1.58i)T+(−26.5+45.8i)T2 1 + (-0.914 - 1.58i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.435−0.754i)T+(−29.5+51.0i)T2 1 + (-0.435 - 0.754i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.66+2.88i)T+(−30.5−52.8i)T2 1 + (-1.66 + 2.88i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.31−5.73i)T+(−33.5+58.0i)T2 1 + (-3.31 - 5.73i)T + (-33.5 + 58.0i)T^{2}
71 1+6.85T+71T2 1 + 6.85T + 71T^{2}
73 1+(1.57+2.72i)T+(−36.5+63.2i)T2 1 + (1.57 + 2.72i)T + (-36.5 + 63.2i)T^{2}
79 1+(−8.78+15.2i)T+(−39.5−68.4i)T2 1 + (-8.78 + 15.2i)T + (-39.5 - 68.4i)T^{2}
83 1−11.4T+83T2 1 - 11.4T + 83T^{2}
89 1+(−0.497+0.861i)T+(−44.5−77.0i)T2 1 + (-0.497 + 0.861i)T + (-44.5 - 77.0i)T^{2}
97 1−13.5T+97T2 1 - 13.5T + 97T^{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

−10.57904239158398380145782526824, −8.912232684722707099112912743873, −8.283516617743011577983515618205, −7.49459175270699025889316682361, −6.62291071188226911183120661230, −5.94737776280281907394558279812, −4.99699894272297430956876460329, −3.17837679678233812712082468460, −2.00341341663886294148050333056, −0.42196071163705207331399610011, 1.98144921099396773280485559571, 3.41146337650272776619838796079, 4.80829074499015603334781662124, 5.25755305664208659623866457265, 6.38001245810156373846997535465, 7.03566171383749781126618250918, 8.720473748186767436066341546719, 9.698910226026898163280617700433, 10.28329980174148899576477878910, 10.67368721610650524040427815802

Graph of the ZZ-function along the critical line