↑

Properties

Label 2-1710-15.2-c1-0-11
Degree 22
Conductor 17101710
Sign 0.501−0.865i0.501 - 0.865i
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 + (3 + 3i)7-s + (0.707 + 0.707i)8-s + (0.999 − 2i)10-s + 1.41i·11-s + (4 − 4i)13-s − 4.24·14-s − 1.00·16-s + (2.82 − 2.82i)17-s − i·19-s + (0.707 + 2.12i)20-s + (−1.00 − 1.00i)22-s + (3.99 − 3i)25-s + 5.65i·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.948 + 0.316i)5-s + (1.13 + 1.13i)7-s + (0.250 + 0.250i)8-s + (0.316 − 0.632i)10-s + 0.426i·11-s + (1.10 − 1.10i)13-s − 1.13·14-s − 0.250·16-s + (0.685 − 0.685i)17-s − 0.229i·19-s + (0.158 + 0.474i)20-s + (−0.213 − 0.213i)22-s + (0.799 − 0.600i)25-s + 1.10i·26-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=((0.501−0.865i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=((0.501−0.865i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: 0.501−0.865i0.501 - 0.865i
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), 0.501−0.865i)(2,\ 1710,\ (\ :1/2),\ 0.501 - 0.865i)

Particular Values

L(1)L(1) ≈\approx 1.3547275131.354727513
L(12)L(\frac12) ≈\approx 1.3547275131.354727513
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+(−3−3i)T+7iT2 1 + (-3 - 3i)T + 7iT^{2}
11 1−1.41iT−11T2 1 - 1.41iT - 11T^{2}
13 1+(−4+4i)T−13iT2 1 + (-4 + 4i)T - 13iT^{2}
17 1+(−2.82+2.82i)T−17iT2 1 + (-2.82 + 2.82i)T - 17iT^{2}
23 1+23iT2 1 + 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+8.48iT−41T2 1 + 8.48iT - 41T^{2}
43 1+(−3+3i)T−43iT2 1 + (-3 + 3i)T - 43iT^{2}
47 1+(5.65−5.65i)T−47iT2 1 + (5.65 - 5.65i)T - 47iT^{2}
53 1+(1.41+1.41i)T+53iT2 1 + (1.41 + 1.41i)T + 53iT^{2}
59 1−5.65T+59T2 1 - 5.65T + 59T^{2}
61 1−10T+61T2 1 - 10T + 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+(3−3i)T−73iT2 1 + (3 - 3i)T - 73iT^{2}
79 1−4iT−79T2 1 - 4iT - 79T^{2}
83 1+(−4.24−4.24i)T+83iT2 1 + (-4.24 - 4.24i)T + 83iT^{2}
89 1−14.1T+89T2 1 - 14.1T + 89T^{2}
97 1+(−12−12i)T+97iT2 1 + (-12 - 12i)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

−9.192551292945969603489670094988, −8.473916251246557979230115512568, −7.966965509853654094111405733310, −7.35874154025630527494548112959, −6.28509923094512289081956411530, −5.41681845215034804006341773331, −4.75506841329842610695331140476, −3.50714607748202021236947555208, −2.40639286137031801168239139888, −0.976531280718952274763813662538, 0.882717726971217881790941326471, 1.69138595461354434651361292456, 3.41959879947052310346672792791, 4.06282301086410843976163526108, 4.71114987316064392578918155711, 6.05708410932538686400310719271, 7.15067284526000728106078053531, 7.82319320373404274762521592516, 8.382535328870710228338613950225, 9.024024233020221669624612273703

Graph of the ZZ-function along the critical line