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Properties

Label 2-640-128.109-c1-0-51
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(621,⋅)\chi_{640} (621, \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

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

Graph of the ZZ-function along the critical line