↑

Properties

Label 2-170-17.8-c1-0-3
Degree 22
Conductor 170170
Sign 0.583+0.811i0.583 + 0.811i
Analytic cond. 1.357451.35745
Root an. cond. 1.165091.16509
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 + (1.10 − 2.66i)3-s − 1.00i·4-s + (0.923 + 0.382i)5-s + (1.10 + 2.66i)6-s + (−0.0470 + 0.0194i)7-s + (0.707 + 0.707i)8-s + (−3.75 − 3.75i)9-s + (−0.923 + 0.382i)10-s + (0.307 + 0.743i)11-s + (−2.66 − 1.10i)12-s − 4.26i·13-s + (0.0194 − 0.0470i)14-s + (2.03 − 2.03i)15-s − 1.00·16-s + (2.43 − 3.32i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.636 − 1.53i)3-s − 0.500i·4-s + (0.413 + 0.171i)5-s + (0.450 + 1.08i)6-s + (−0.0177 + 0.00736i)7-s + (0.250 + 0.250i)8-s + (−1.25 − 1.25i)9-s + (−0.292 + 0.121i)10-s + (0.0928 + 0.224i)11-s + (−0.768 − 0.318i)12-s − 1.18i·13-s + (0.00521 − 0.0125i)14-s + (0.526 − 0.526i)15-s − 0.250·16-s + (0.589 − 0.807i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.583+0.811i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.583+0.811i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.583+0.811i0.583 + 0.811i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(161,⋅)\chi_{170} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.583+0.811i)(2,\ 170,\ (\ :1/2),\ 0.583 + 0.811i)

Particular Values

L(1)L(1) ≈\approx 1.00378−0.514592i1.00378 - 0.514592i
L(12)L(\frac12) ≈\approx 1.00378−0.514592i1.00378 - 0.514592i
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
5 1+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)T
17 1+(−2.43+3.32i)T 1 + (-2.43 + 3.32i)T
good3 1+(−1.10+2.66i)T+(−2.12−2.12i)T2 1 + (-1.10 + 2.66i)T + (-2.12 - 2.12i)T^{2}
7 1+(0.0470−0.0194i)T+(4.94−4.94i)T2 1 + (0.0470 - 0.0194i)T + (4.94 - 4.94i)T^{2}
11 1+(−0.307−0.743i)T+(−7.77+7.77i)T2 1 + (-0.307 - 0.743i)T + (-7.77 + 7.77i)T^{2}
13 1+4.26iT−13T2 1 + 4.26iT - 13T^{2}
19 1+(4.53−4.53i)T−19iT2 1 + (4.53 - 4.53i)T - 19iT^{2}
23 1+(−3.28−7.94i)T+(−16.2+16.2i)T2 1 + (-3.28 - 7.94i)T + (-16.2 + 16.2i)T^{2}
29 1+(−4.22−1.74i)T+(20.5+20.5i)T2 1 + (-4.22 - 1.74i)T + (20.5 + 20.5i)T^{2}
31 1+(0.189−0.456i)T+(−21.9−21.9i)T2 1 + (0.189 - 0.456i)T + (-21.9 - 21.9i)T^{2}
37 1+(−0.598+1.44i)T+(−26.1−26.1i)T2 1 + (-0.598 + 1.44i)T + (-26.1 - 26.1i)T^{2}
41 1+(7.95−3.29i)T+(28.9−28.9i)T2 1 + (7.95 - 3.29i)T + (28.9 - 28.9i)T^{2}
43 1+(−1.73−1.73i)T+43iT2 1 + (-1.73 - 1.73i)T + 43iT^{2}
47 1−8.22iT−47T2 1 - 8.22iT - 47T^{2}
53 1+(−7.79+7.79i)T−53iT2 1 + (-7.79 + 7.79i)T - 53iT^{2}
59 1+(−7.13−7.13i)T+59iT2 1 + (-7.13 - 7.13i)T + 59iT^{2}
61 1+(5.73−2.37i)T+(43.1−43.1i)T2 1 + (5.73 - 2.37i)T + (43.1 - 43.1i)T^{2}
67 1+0.822T+67T2 1 + 0.822T + 67T^{2}
71 1+(−4.17+10.0i)T+(−50.2−50.2i)T2 1 + (-4.17 + 10.0i)T + (-50.2 - 50.2i)T^{2}
73 1+(6.19+2.56i)T+(51.6+51.6i)T2 1 + (6.19 + 2.56i)T + (51.6 + 51.6i)T^{2}
79 1+(3.14+7.59i)T+(−55.8+55.8i)T2 1 + (3.14 + 7.59i)T + (-55.8 + 55.8i)T^{2}
83 1+(0.955−0.955i)T−83iT2 1 + (0.955 - 0.955i)T - 83iT^{2}
89 1−17.0iT−89T2 1 - 17.0iT - 89T^{2}
97 1+(−2.95−1.22i)T+(68.5+68.5i)T2 1 + (-2.95 - 1.22i)T + (68.5 + 68.5i)T^{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

−12.84605075232476612125572682832, −11.86456058815288171218458581910, −10.43796584817495533204618237238, −9.332936304217948410965715036238, −8.196699789367354289431387810666, −7.53684221611807908743618417985, −6.56775087292169688244473827815, −5.50708719406736682725578301598, −2.97996733739649117076777464069, −1.41454334829303442353464486078, 2.45527571290702436286840066973, 3.89767838821113505100559978046, 4.85370249421076868733338516836, 6.63229003810346678034683534522, 8.601939505941634414010430365802, 8.814017999416134368525780270385, 10.01781133679447388301152929318, 10.54780970509608502960671945735, 11.60105422447455780538154578291, 12.91364911305399727429821984824

Graph of the ZZ-function along the critical line