↑

Properties

Label 2-538-269.11-c1-0-6
Degree 22
Conductor 538538
Sign 0.664−0.747i0.664 - 0.747i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
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  + (−0.902 + 0.430i)2-s + (−0.405 + 2.86i)3-s + (0.628 − 0.777i)4-s + (0.0702 − 2.99i)5-s + (−0.867 − 2.75i)6-s + (−1.94 − 2.77i)7-s + (−0.232 + 0.972i)8-s + (−5.14 − 1.48i)9-s + (1.22 + 2.73i)10-s + (1.08 + 2.94i)11-s + (1.97 + 2.11i)12-s + (0.409 − 0.183i)13-s + (2.94 + 1.67i)14-s + (8.55 + 1.41i)15-s + (−0.209 − 0.977i)16-s + (7.08 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.638 + 0.304i)2-s + (−0.233 + 1.65i)3-s + (0.314 − 0.388i)4-s + (0.0314 − 1.34i)5-s + (−0.354 − 1.12i)6-s + (−0.733 − 1.05i)7-s + (−0.0821 + 0.343i)8-s + (−1.71 − 0.495i)9-s + (0.388 + 0.864i)10-s + (0.325 + 0.887i)11-s + (0.568 + 0.610i)12-s + (0.113 − 0.0509i)13-s + (0.788 + 0.446i)14-s + (2.20 + 0.365i)15-s + (−0.0523 − 0.244i)16-s + (1.71 + 0.630i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.664−0.747i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.664−0.747i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.664−0.747i0.664 - 0.747i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(11,⋅)\chi_{538} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 0.664−0.747i)(2,\ 538,\ (\ :1/2),\ 0.664 - 0.747i)

Particular Values

L(1)L(1) ≈\approx 0.856146+0.384564i0.856146 + 0.384564i
L(12)L(\frac12) ≈\approx 0.856146+0.384564i0.856146 + 0.384564i
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+(0.902−0.430i)T 1 + (0.902 - 0.430i)T
269 1+(11.4−11.7i)T 1 + (11.4 - 11.7i)T
good3 1+(0.405−2.86i)T+(−2.88−0.832i)T2 1 + (0.405 - 2.86i)T + (-2.88 - 0.832i)T^{2}
5 1+(−0.0702+2.99i)T+(−4.99−0.234i)T2 1 + (-0.0702 + 2.99i)T + (-4.99 - 0.234i)T^{2}
7 1+(1.94+2.77i)T+(−2.41+6.57i)T2 1 + (1.94 + 2.77i)T + (-2.41 + 6.57i)T^{2}
11 1+(−1.08−2.94i)T+(−8.38+7.11i)T2 1 + (-1.08 - 2.94i)T + (-8.38 + 7.11i)T^{2}
13 1+(−0.409+0.183i)T+(8.63−9.71i)T2 1 + (-0.409 + 0.183i)T + (8.63 - 9.71i)T^{2}
17 1+(−7.08−2.59i)T+(12.9+10.9i)T2 1 + (-7.08 - 2.59i)T + (12.9 + 10.9i)T^{2}
19 1+(−0.861−2.18i)T+(−13.8+12.9i)T2 1 + (-0.861 - 2.18i)T + (-13.8 + 12.9i)T^{2}
23 1+(−4.82−0.683i)T+(22.0+6.38i)T2 1 + (-4.82 - 0.683i)T + (22.0 + 6.38i)T^{2}
29 1+(0.421+8.98i)T+(−28.8+2.71i)T2 1 + (0.421 + 8.98i)T + (-28.8 + 2.71i)T^{2}
31 1+(−6.02−7.10i)T+(−5.06+30.5i)T2 1 + (-6.02 - 7.10i)T + (-5.06 + 30.5i)T^{2}
37 1+(−7.15−1.35i)T+(34.4+13.5i)T2 1 + (-7.15 - 1.35i)T + (34.4 + 13.5i)T^{2}
41 1+(0.866−1.81i)T+(−25.7−31.8i)T2 1 + (0.866 - 1.81i)T + (-25.7 - 31.8i)T^{2}
43 1+(−7.44+0.349i)T+(42.8−4.02i)T2 1 + (-7.44 + 0.349i)T + (42.8 - 4.02i)T^{2}
47 1+(1.49+12.6i)T+(−45.7+10.9i)T2 1 + (1.49 + 12.6i)T + (-45.7 + 10.9i)T^{2}
53 1+(5.03+3.87i)T+(13.5+51.2i)T2 1 + (5.03 + 3.87i)T + (13.5 + 51.2i)T^{2}
59 1+(1.90+1.54i)T+(12.3+57.6i)T2 1 + (1.90 + 1.54i)T + (12.3 + 57.6i)T^{2}
61 1+(−1.19−1.62i)T+(−18.3+58.1i)T2 1 + (-1.19 - 1.62i)T + (-18.3 + 58.1i)T^{2}
67 1+(−8.10−10.0i)T+(−14.0+65.5i)T2 1 + (-8.10 - 10.0i)T + (-14.0 + 65.5i)T^{2}
71 1+(5.52+9.75i)T+(−36.4+60.9i)T2 1 + (5.52 + 9.75i)T + (-36.4 + 60.9i)T^{2}
73 1+(7.15+11.9i)T+(−34.5+64.3i)T2 1 + (7.15 + 11.9i)T + (-34.5 + 64.3i)T^{2}
79 1+(4.91−5.03i)T+(−1.85−78.9i)T2 1 + (4.91 - 5.03i)T + (-1.85 - 78.9i)T^{2}
83 1+(−3.41+5.41i)T+(−35.7−74.9i)T2 1 + (-3.41 + 5.41i)T + (-35.7 - 74.9i)T^{2}
89 1+(−0.872−1.31i)T+(−34.5+82.0i)T2 1 + (-0.872 - 1.31i)T + (-34.5 + 82.0i)T^{2}
97 1+(2.62−3.58i)T+(−29.1−92.5i)T2 1 + (2.62 - 3.58i)T + (-29.1 - 92.5i)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

−10.37146671425770804544153192474, −9.980524456775106096068166693647, −9.456300637611862297530477544760, −8.530621025568135965169379986863, −7.57185433950153166896400493598, −6.20776037683552190002136282743, −5.19154130192810073104608006671, −4.38762814119296475806555173937, −3.47196390143824745895288148910, −0.978486862162350204958044079531, 1.03101214419777374341071785535, 2.78328852925823401385107564825, 2.97975167047838906019203230146, 5.75056135018102675459808790635, 6.31636033032643748331067752824, 7.15086616353671066646105123258, 7.80852737861204435754952244662, 8.899176269669940302279325129308, 9.757669313603721499964352155733, 11.05755437655372668370218740637

Graph of the ZZ-function along the critical line