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Properties

Label 2-640-128.101-c1-0-19
Degree 22
Conductor 640640
Sign 0.832−0.553i0.832 - 0.553i
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  + (0.810 − 1.15i)2-s + (−1.01 + 0.542i)3-s + (−0.685 − 1.87i)4-s + (0.634 + 0.773i)5-s + (−0.194 + 1.61i)6-s + (−1.73 + 1.15i)7-s + (−2.73 − 0.728i)8-s + (−0.930 + 1.39i)9-s + (1.41 − 0.108i)10-s + (−0.0953 + 0.0289i)11-s + (1.71 + 1.53i)12-s + (5.37 + 4.40i)13-s + (−0.0631 + 2.94i)14-s + (−1.06 − 0.440i)15-s + (−3.06 + 2.57i)16-s + (2.51 − 1.04i)17-s + ⋯
L(s)  = 1  + (0.573 − 0.819i)2-s + (−0.586 + 0.313i)3-s + (−0.342 − 0.939i)4-s + (0.283 + 0.345i)5-s + (−0.0792 + 0.659i)6-s + (−0.655 + 0.438i)7-s + (−0.966 − 0.257i)8-s + (−0.310 + 0.464i)9-s + (0.445 − 0.0342i)10-s + (−0.0287 + 0.00871i)11-s + (0.495 + 0.443i)12-s + (1.48 + 1.22i)13-s + (−0.0168 + 0.788i)14-s + (−0.274 − 0.113i)15-s + (−0.765 + 0.644i)16-s + (0.609 − 0.252i)17-s + ⋯

Functional equation

Λ(s)=(640s/2 ΓC(s) L(s)=((0.832−0.553i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(640s/2 ΓC(s+1/2) L(s)=((0.832−0.553i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: 0.832−0.553i0.832 - 0.553i
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), 0.832−0.553i)(2,\ 640,\ (\ :1/2),\ 0.832 - 0.553i)

Particular Values

L(1)L(1) ≈\approx 1.24949+0.377234i1.24949 + 0.377234i
L(12)L(\frac12) ≈\approx 1.24949+0.377234i1.24949 + 0.377234i
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+(−0.810+1.15i)T 1 + (-0.810 + 1.15i)T
5 1+(−0.634−0.773i)T 1 + (-0.634 - 0.773i)T
good3 1+(1.01−0.542i)T+(1.66−2.49i)T2 1 + (1.01 - 0.542i)T + (1.66 - 2.49i)T^{2}
7 1+(1.73−1.15i)T+(2.67−6.46i)T2 1 + (1.73 - 1.15i)T + (2.67 - 6.46i)T^{2}
11 1+(0.0953−0.0289i)T+(9.14−6.11i)T2 1 + (0.0953 - 0.0289i)T + (9.14 - 6.11i)T^{2}
13 1+(−5.37−4.40i)T+(2.53+12.7i)T2 1 + (-5.37 - 4.40i)T + (2.53 + 12.7i)T^{2}
17 1+(−2.51+1.04i)T+(12.0−12.0i)T2 1 + (-2.51 + 1.04i)T + (12.0 - 12.0i)T^{2}
19 1+(−0.383−3.89i)T+(−18.6+3.70i)T2 1 + (-0.383 - 3.89i)T + (-18.6 + 3.70i)T^{2}
23 1+(3.00−0.597i)T+(21.2−8.80i)T2 1 + (3.00 - 0.597i)T + (21.2 - 8.80i)T^{2}
29 1+(0.430−1.41i)T+(−24.1−16.1i)T2 1 + (0.430 - 1.41i)T + (-24.1 - 16.1i)T^{2}
31 1+(−5.84−5.84i)T+31iT2 1 + (-5.84 - 5.84i)T + 31iT^{2}
37 1+(2.21+0.217i)T+(36.2+7.21i)T2 1 + (2.21 + 0.217i)T + (36.2 + 7.21i)T^{2}
41 1+(−0.994−5.00i)T+(−37.8+15.6i)T2 1 + (-0.994 - 5.00i)T + (-37.8 + 15.6i)T^{2}
43 1+(1.07+0.575i)T+(23.8+35.7i)T2 1 + (1.07 + 0.575i)T + (23.8 + 35.7i)T^{2}
47 1+(2.33+5.64i)T+(−33.2+33.2i)T2 1 + (2.33 + 5.64i)T + (-33.2 + 33.2i)T^{2}
53 1+(−0.222−0.735i)T+(−44.0+29.4i)T2 1 + (-0.222 - 0.735i)T + (-44.0 + 29.4i)T^{2}
59 1+(1.01−0.834i)T+(11.5−57.8i)T2 1 + (1.01 - 0.834i)T + (11.5 - 57.8i)T^{2}
61 1+(0.862+1.61i)T+(−33.8+50.7i)T2 1 + (0.862 + 1.61i)T + (-33.8 + 50.7i)T^{2}
67 1+(5.00+9.36i)T+(−37.2+55.7i)T2 1 + (5.00 + 9.36i)T + (-37.2 + 55.7i)T^{2}
71 1+(−5.05−7.56i)T+(−27.1+65.5i)T2 1 + (-5.05 - 7.56i)T + (-27.1 + 65.5i)T^{2}
73 1+(−5.49−3.67i)T+(27.9+67.4i)T2 1 + (-5.49 - 3.67i)T + (27.9 + 67.4i)T^{2}
79 1+(−2.09+5.06i)T+(−55.8−55.8i)T2 1 + (-2.09 + 5.06i)T + (-55.8 - 55.8i)T^{2}
83 1+(15.0−1.48i)T+(81.4−16.1i)T2 1 + (15.0 - 1.48i)T + (81.4 - 16.1i)T^{2}
89 1+(−16.6−3.32i)T+(82.2+34.0i)T2 1 + (-16.6 - 3.32i)T + (82.2 + 34.0i)T^{2}
97 1+(1.75+1.75i)T+97iT2 1 + (1.75 + 1.75i)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

−10.70480417095886657836412210777, −10.08217784549266112877502887440, −9.243536216453658149083521763566, −8.276825171188891440778533131956, −6.54855652653108119253120387214, −6.02937026288434087591531708453, −5.15196300984626270074787929381, −3.98414766739014511395968669312, −3.00649930559450576097158803600, −1.63896042412214901026510654484, 0.65944206455715671137956963354, 3.06336068583343485566466627311, 3.97574173222806785981851880152, 5.32919027667094336240936447115, 6.05305873500612835916192502446, 6.57053569823920268681155373012, 7.74852436086061859810682514203, 8.536762938447101420382479227955, 9.482755452473597566984167270328, 10.55656981000154771360639880357

Graph of the ZZ-function along the critical line