↑

Properties

Label 2-510-15.2-c1-0-3
Degree 22
Conductor 510510
Sign 0.526−0.850i0.526 - 0.850i
Analytic cond. 4.072374.07237
Root an. cond. 2.018012.01801
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.58 + 0.705i)3-s − 1.00i·4-s + (−2.23 + 0.0974i)5-s + (0.619 − 1.61i)6-s + (−3.25 − 3.25i)7-s + (0.707 + 0.707i)8-s + (2.00 − 2.23i)9-s + (1.51 − 1.64i)10-s + 3.90i·11-s + (0.705 + 1.58i)12-s + (2.89 − 2.89i)13-s + 4.60·14-s + (3.46 − 1.73i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.913 + 0.407i)3-s − 0.500i·4-s + (−0.999 + 0.0435i)5-s + (0.252 − 0.660i)6-s + (−1.23 − 1.23i)7-s + (0.250 + 0.250i)8-s + (0.668 − 0.744i)9-s + (0.477 − 0.521i)10-s + 1.17i·11-s + (0.203 + 0.456i)12-s + (0.802 − 0.802i)13-s + 1.23·14-s + (0.894 − 0.446i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯

Functional equation

Λ(s)=(510s/2 ΓC(s) L(s)=((0.526−0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(510s/2 ΓC(s+1/2) L(s)=((0.526−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 510510    =    2⋅3⋅5⋅172 \cdot 3 \cdot 5 \cdot 17
Sign: 0.526−0.850i0.526 - 0.850i
Analytic conductor: 4.072374.07237
Root analytic conductor: 2.018012.01801
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ510(137,⋅)\chi_{510} (137, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 510, ( :1/2), 0.526−0.850i)(2,\ 510,\ (\ :1/2),\ 0.526 - 0.850i)

Particular Values

L(1)L(1) ≈\approx 0.430624+0.239969i0.430624 + 0.239969i
L(12)L(\frac12) ≈\approx 0.430624+0.239969i0.430624 + 0.239969i
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1+(1.58−0.705i)T 1 + (1.58 - 0.705i)T
5 1+(2.23−0.0974i)T 1 + (2.23 - 0.0974i)T
17 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good7 1+(3.25+3.25i)T+7iT2 1 + (3.25 + 3.25i)T + 7iT^{2}
11 1−3.90iT−11T2 1 - 3.90iT - 11T^{2}
13 1+(−2.89+2.89i)T−13iT2 1 + (-2.89 + 2.89i)T - 13iT^{2}
19 1−5.28iT−19T2 1 - 5.28iT - 19T^{2}
23 1+(−1.08−1.08i)T+23iT2 1 + (-1.08 - 1.08i)T + 23iT^{2}
29 1+1.81T+29T2 1 + 1.81T + 29T^{2}
31 1−7.47T+31T2 1 - 7.47T + 31T^{2}
37 1+(3.81+3.81i)T+37iT2 1 + (3.81 + 3.81i)T + 37iT^{2}
41 1+3.03iT−41T2 1 + 3.03iT - 41T^{2}
43 1+(3.66−3.66i)T−43iT2 1 + (3.66 - 3.66i)T - 43iT^{2}
47 1+(−3.79+3.79i)T−47iT2 1 + (-3.79 + 3.79i)T - 47iT^{2}
53 1+(−8.80−8.80i)T+53iT2 1 + (-8.80 - 8.80i)T + 53iT^{2}
59 1−8.47T+59T2 1 - 8.47T + 59T^{2}
61 1−10.1T+61T2 1 - 10.1T + 61T^{2}
67 1+(−4.29−4.29i)T+67iT2 1 + (-4.29 - 4.29i)T + 67iT^{2}
71 1−3.19iT−71T2 1 - 3.19iT - 71T^{2}
73 1+(−7.14+7.14i)T−73iT2 1 + (-7.14 + 7.14i)T - 73iT^{2}
79 1+3.97iT−79T2 1 + 3.97iT - 79T^{2}
83 1+(−3.62−3.62i)T+83iT2 1 + (-3.62 - 3.62i)T + 83iT^{2}
89 1+7.98T+89T2 1 + 7.98T + 89T^{2}
97 1+(−9.40−9.40i)T+97iT2 1 + (-9.40 - 9.40i)T + 97iT^{2}
show more
show less
   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

−10.62063262399240875021292599038, −10.33211832188666828537263361722, −9.504983733754558738719573598690, −8.201751864414330495381204084879, −7.20308758341980175521611799749, −6.70540063319784072508430837271, −5.60524640352090101098994102279, −4.26627134089209604858274498242, −3.60277955450784797593376067841, −0.812951138505086513704700070289, 0.61528929760268119891162084348, 2.61871055434685051003336613880, 3.75391900985423541395985075046, 5.16361424678102267760192431016, 6.38765918920632002460551808969, 6.90129249124186279233130779770, 8.380445371880188960053866610264, 8.842365713011215166601565564343, 9.952467628634500801911774043503, 11.12384346288352627858323963081

Graph of the ZZ-function along the critical line