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Properties

Label 2-640-128.101-c1-0-1
Degree 22
Conductor 640640
Sign −0.467+0.883i-0.467 + 0.883i
Analytic cond. 5.110425.11042
Root an. cond. 2.260622.26062
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  + (−0.726 + 1.21i)2-s + (−1.23 + 0.662i)3-s + (−0.944 − 1.76i)4-s + (0.634 + 0.773i)5-s + (0.0963 − 1.98i)6-s + (0.491 − 0.328i)7-s + (2.82 + 0.133i)8-s + (−0.568 + 0.850i)9-s + (−1.39 + 0.208i)10-s + (1.28 − 0.390i)11-s + (2.33 + 1.55i)12-s + (−3.74 − 3.07i)13-s + (0.0415 + 0.834i)14-s + (−1.29 − 0.538i)15-s + (−2.21 + 3.33i)16-s + (−4.56 + 1.89i)17-s + ⋯
L(s)  = 1  + (−0.513 + 0.858i)2-s + (−0.715 + 0.382i)3-s + (−0.472 − 0.881i)4-s + (0.283 + 0.345i)5-s + (0.0393 − 0.810i)6-s + (0.185 − 0.124i)7-s + (0.998 + 0.0472i)8-s + (−0.189 + 0.283i)9-s + (−0.442 + 0.0658i)10-s + (0.388 − 0.117i)11-s + (0.675 + 0.450i)12-s + (−1.03 − 0.852i)13-s + (0.0110 + 0.223i)14-s + (−0.335 − 0.138i)15-s + (−0.553 + 0.832i)16-s + (−1.10 + 0.458i)17-s + ⋯

Functional equation

Λ(s)=(640s/2 ΓC(s) L(s)=((−0.467+0.883i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(640s/2 ΓC(s+1/2) L(s)=((−0.467+0.883i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: −0.467+0.883i-0.467 + 0.883i
Analytic conductor: 5.110425.11042
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ640(101,⋅)\chi_{640} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 640, ( :1/2), −0.467+0.883i)(2,\ 640,\ (\ :1/2),\ -0.467 + 0.883i)

Particular Values

L(1)L(1) ≈\approx 0.0678491−0.112669i0.0678491 - 0.112669i
L(12)L(\frac12) ≈\approx 0.0678491−0.112669i0.0678491 - 0.112669i
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)
bad2 1+(0.726−1.21i)T 1 + (0.726 - 1.21i)T
5 1+(−0.634−0.773i)T 1 + (-0.634 - 0.773i)T
good3 1+(1.23−0.662i)T+(1.66−2.49i)T2 1 + (1.23 - 0.662i)T + (1.66 - 2.49i)T^{2}
7 1+(−0.491+0.328i)T+(2.67−6.46i)T2 1 + (-0.491 + 0.328i)T + (2.67 - 6.46i)T^{2}
11 1+(−1.28+0.390i)T+(9.14−6.11i)T2 1 + (-1.28 + 0.390i)T + (9.14 - 6.11i)T^{2}
13 1+(3.74+3.07i)T+(2.53+12.7i)T2 1 + (3.74 + 3.07i)T + (2.53 + 12.7i)T^{2}
17 1+(4.56−1.89i)T+(12.0−12.0i)T2 1 + (4.56 - 1.89i)T + (12.0 - 12.0i)T^{2}
19 1+(−0.748−7.59i)T+(−18.6+3.70i)T2 1 + (-0.748 - 7.59i)T + (-18.6 + 3.70i)T^{2}
23 1+(3.22−0.642i)T+(21.2−8.80i)T2 1 + (3.22 - 0.642i)T + (21.2 - 8.80i)T^{2}
29 1+(−1.38+4.55i)T+(−24.1−16.1i)T2 1 + (-1.38 + 4.55i)T + (-24.1 - 16.1i)T^{2}
31 1+(0.603+0.603i)T+31iT2 1 + (0.603 + 0.603i)T + 31iT^{2}
37 1+(10.1+0.999i)T+(36.2+7.21i)T2 1 + (10.1 + 0.999i)T + (36.2 + 7.21i)T^{2}
41 1+(2.10+10.5i)T+(−37.8+15.6i)T2 1 + (2.10 + 10.5i)T + (-37.8 + 15.6i)T^{2}
43 1+(2.32+1.24i)T+(23.8+35.7i)T2 1 + (2.32 + 1.24i)T + (23.8 + 35.7i)T^{2}
47 1+(−3.11−7.52i)T+(−33.2+33.2i)T2 1 + (-3.11 - 7.52i)T + (-33.2 + 33.2i)T^{2}
53 1+(2.45+8.10i)T+(−44.0+29.4i)T2 1 + (2.45 + 8.10i)T + (-44.0 + 29.4i)T^{2}
59 1+(−3.24+2.65i)T+(11.5−57.8i)T2 1 + (-3.24 + 2.65i)T + (11.5 - 57.8i)T^{2}
61 1+(3.50+6.56i)T+(−33.8+50.7i)T2 1 + (3.50 + 6.56i)T + (-33.8 + 50.7i)T^{2}
67 1+(2.96+5.54i)T+(−37.2+55.7i)T2 1 + (2.96 + 5.54i)T + (-37.2 + 55.7i)T^{2}
71 1+(0.235+0.353i)T+(−27.1+65.5i)T2 1 + (0.235 + 0.353i)T + (-27.1 + 65.5i)T^{2}
73 1+(−5.36−3.58i)T+(27.9+67.4i)T2 1 + (-5.36 - 3.58i)T + (27.9 + 67.4i)T^{2}
79 1+(5.03−12.1i)T+(−55.8−55.8i)T2 1 + (5.03 - 12.1i)T + (-55.8 - 55.8i)T^{2}
83 1+(5.72−0.563i)T+(81.4−16.1i)T2 1 + (5.72 - 0.563i)T + (81.4 - 16.1i)T^{2}
89 1+(11.5+2.29i)T+(82.2+34.0i)T2 1 + (11.5 + 2.29i)T + (82.2 + 34.0i)T^{2}
97 1+(−11.4−11.4i)T+97iT2 1 + (-11.4 - 11.4i)T + 97iT^{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

−10.72897608431912757178789624143, −10.31514854510217396477083510278, −9.553237618461870276661525122179, −8.362408178718593608167303583283, −7.70874019929076029550518488019, −6.60705882839828204942481387597, −5.79304260731559426628129461738, −5.11837258371480291578513585663, −3.97455112719891226674928554772, −2.00105791854458422748566614855, 0.090100458077449491605872925145, 1.64609389713097040880774430690, 2.84981566927584436335609728100, 4.44072798116961355125387203336, 5.13891182665617830344826420794, 6.69102700402496921638885125367, 7.15813009955398935085324824692, 8.704921526851681171072242313483, 9.085098782049047997683885341988, 10.00351359510391104378748283283

Graph of the ZZ-function along the critical line