↑

Properties

Label 2-1710-15.2-c1-0-25
Degree 22
Conductor 17101710
Sign 0.0618+0.998i0.0618 + 0.998i
Analytic cond. 13.654413.6544
Root an. cond. 3.695183.69518
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 − 1.00i·4-s + (2.12 + 0.707i)5-s + (−1 − i)7-s + (−0.707 − 0.707i)8-s + (2 − 0.999i)10-s − 1.41i·11-s − 1.41·14-s − 1.00·16-s + (4.24 − 4.24i)17-s − i·19-s + (0.707 − 2.12i)20-s + (−1.00 − 1.00i)22-s + (1.41 + 1.41i)23-s + (3.99 + 3i)25-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.948 + 0.316i)5-s + (−0.377 − 0.377i)7-s + (−0.250 − 0.250i)8-s + (0.632 − 0.316i)10-s − 0.426i·11-s − 0.377·14-s − 0.250·16-s + (1.02 − 1.02i)17-s − 0.229i·19-s + (0.158 − 0.474i)20-s + (−0.213 − 0.213i)22-s + (0.294 + 0.294i)23-s + (0.799 + 0.600i)25-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=((0.0618+0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=((0.0618+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: 0.0618+0.998i0.0618 + 0.998i
Analytic conductor: 13.654413.6544
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1710(647,⋅)\chi_{1710} (647, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1710, ( :1/2), 0.0618+0.998i)(2,\ 1710,\ (\ :1/2),\ 0.0618 + 0.998i)

Particular Values

L(1)L(1) ≈\approx 2.5381701312.538170131
L(12)L(\frac12) ≈\approx 2.5381701312.538170131
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
5 1+(−2.12−0.707i)T 1 + (-2.12 - 0.707i)T
19 1+iT 1 + iT
good7 1+(1+i)T+7iT2 1 + (1 + i)T + 7iT^{2}
11 1+1.41iT−11T2 1 + 1.41iT - 11T^{2}
13 1−13iT2 1 - 13iT^{2}
17 1+(−4.24+4.24i)T−17iT2 1 + (-4.24 + 4.24i)T - 17iT^{2}
23 1+(−1.41−1.41i)T+23iT2 1 + (-1.41 - 1.41i)T + 23iT^{2}
29 1+2.82T+29T2 1 + 2.82T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+(6+6i)T+37iT2 1 + (6 + 6i)T + 37iT^{2}
41 1+11.3iT−41T2 1 + 11.3iT - 41T^{2}
43 1+(−3+3i)T−43iT2 1 + (-3 + 3i)T - 43iT^{2}
47 1+(7.07−7.07i)T−47iT2 1 + (7.07 - 7.07i)T - 47iT^{2}
53 1+(−1.41−1.41i)T+53iT2 1 + (-1.41 - 1.41i)T + 53iT^{2}
59 1−2.82T+59T2 1 - 2.82T + 59T^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1+(−10−10i)T+67iT2 1 + (-10 - 10i)T + 67iT^{2}
71 1−5.65iT−71T2 1 - 5.65iT - 71T^{2}
73 1+(−9+9i)T−73iT2 1 + (-9 + 9i)T - 73iT^{2}
79 1−8iT−79T2 1 - 8iT - 79T^{2}
83 1+(8.48+8.48i)T+83iT2 1 + (8.48 + 8.48i)T + 83iT^{2}
89 1+11.3T+89T2 1 + 11.3T + 89T^{2}
97 1+(2+2i)T+97iT2 1 + (2 + 2i)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

−9.456599040807511140730493778546, −8.539127281055136789539518396463, −7.25845146536437158454766347716, −6.74152729175077164329176548004, −5.59054812089198710346927697970, −5.28560865616001553825914489152, −3.91289098322957608326858064397, −3.10512220287165935691542739355, −2.19032876562780736365635104749, −0.858124759171857129207347469440, 1.47792363691069479629498611550, 2.66256432474343768382093817764, 3.67350454609545778902984369394, 4.81610720643524675242852432074, 5.48913057530040719556746272320, 6.27362563583692284590641822610, 6.83682133418243437488109277221, 8.065799252071138023024565827372, 8.533601358640846305929352890154, 9.713070339289918673006798671498

Graph of the ZZ-function along the critical line