↑

Properties

Label 2-116-116.15-c1-0-5
Degree 22
Conductor 116116
Sign 0.372−0.928i0.372 - 0.928i
Analytic cond. 0.9262640.926264
Root an. cond. 0.9624260.962426
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  + (1.03 + 0.961i)2-s + (0.381 + 0.606i)3-s + (0.151 + 1.99i)4-s + (−0.388 − 0.309i)5-s + (−0.187 + 0.996i)6-s + (−1.45 − 0.331i)7-s + (−1.76 + 2.21i)8-s + (1.07 − 2.24i)9-s + (−0.105 − 0.694i)10-s + (1.07 − 3.08i)11-s + (−1.15 + 0.852i)12-s + (0.610 + 1.26i)13-s + (−1.18 − 1.73i)14-s + (0.0398 − 0.353i)15-s + (−3.95 + 0.604i)16-s + (2.80 + 2.80i)17-s + ⋯
L(s)  = 1  + (0.733 + 0.679i)2-s + (0.220 + 0.350i)3-s + (0.0757 + 0.997i)4-s + (−0.173 − 0.138i)5-s + (−0.0767 + 0.406i)6-s + (−0.548 − 0.125i)7-s + (−0.622 + 0.782i)8-s + (0.359 − 0.746i)9-s + (−0.0332 − 0.219i)10-s + (0.325 − 0.929i)11-s + (−0.332 + 0.246i)12-s + (0.169 + 0.351i)13-s + (−0.316 − 0.464i)14-s + (0.0102 − 0.0913i)15-s + (−0.988 + 0.151i)16-s + (0.679 + 0.679i)17-s + ⋯

Functional equation

Λ(s)=(116s/2 ΓC(s) L(s)=((0.372−0.928i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(116s/2 ΓC(s+1/2) L(s)=((0.372−0.928i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 116116    =    22⋅292^{2} \cdot 29
Sign: 0.372−0.928i0.372 - 0.928i
Analytic conductor: 0.9262640.926264
Root analytic conductor: 0.9624260.962426
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ116(15,⋅)\chi_{116} (15, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 116, ( :1/2), 0.372−0.928i)(2,\ 116,\ (\ :1/2),\ 0.372 - 0.928i)

Particular Values

L(1)L(1) ≈\approx 1.22670+0.829745i1.22670 + 0.829745i
L(12)L(\frac12) ≈\approx 1.22670+0.829745i1.22670 + 0.829745i
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+(−1.03−0.961i)T 1 + (-1.03 - 0.961i)T
29 1+(3.77−3.84i)T 1 + (3.77 - 3.84i)T
good3 1+(−0.381−0.606i)T+(−1.30+2.70i)T2 1 + (-0.381 - 0.606i)T + (-1.30 + 2.70i)T^{2}
5 1+(0.388+0.309i)T+(1.11+4.87i)T2 1 + (0.388 + 0.309i)T + (1.11 + 4.87i)T^{2}
7 1+(1.45+0.331i)T+(6.30+3.03i)T2 1 + (1.45 + 0.331i)T + (6.30 + 3.03i)T^{2}
11 1+(−1.07+3.08i)T+(−8.60−6.85i)T2 1 + (-1.07 + 3.08i)T + (-8.60 - 6.85i)T^{2}
13 1+(−0.610−1.26i)T+(−8.10+10.1i)T2 1 + (-0.610 - 1.26i)T + (-8.10 + 10.1i)T^{2}
17 1+(−2.80−2.80i)T+17iT2 1 + (-2.80 - 2.80i)T + 17iT^{2}
19 1+(4.35+2.73i)T+(8.24+17.1i)T2 1 + (4.35 + 2.73i)T + (8.24 + 17.1i)T^{2}
23 1+(−0.0143+0.0114i)T+(5.11−22.4i)T2 1 + (-0.0143 + 0.0114i)T + (5.11 - 22.4i)T^{2}
31 1+(−0.262−2.32i)T+(−30.2+6.89i)T2 1 + (-0.262 - 2.32i)T + (-30.2 + 6.89i)T^{2}
37 1+(1.69−0.592i)T+(28.9−23.0i)T2 1 + (1.69 - 0.592i)T + (28.9 - 23.0i)T^{2}
41 1+(0.378−0.378i)T−41iT2 1 + (0.378 - 0.378i)T - 41iT^{2}
43 1+(3.80+0.429i)T+(41.9+9.56i)T2 1 + (3.80 + 0.429i)T + (41.9 + 9.56i)T^{2}
47 1+(−8.34−2.92i)T+(36.7+29.3i)T2 1 + (-8.34 - 2.92i)T + (36.7 + 29.3i)T^{2}
53 1+(6.44−8.07i)T+(−11.7−51.6i)T2 1 + (6.44 - 8.07i)T + (-11.7 - 51.6i)T^{2}
59 1−14.5iT−59T2 1 - 14.5iT - 59T^{2}
61 1+(−0.184+0.116i)T+(26.4−54.9i)T2 1 + (-0.184 + 0.116i)T + (26.4 - 54.9i)T^{2}
67 1+(−1.44−0.695i)T+(41.7+52.3i)T2 1 + (-1.44 - 0.695i)T + (41.7 + 52.3i)T^{2}
71 1+(−11.0+5.30i)T+(44.2−55.5i)T2 1 + (-11.0 + 5.30i)T + (44.2 - 55.5i)T^{2}
73 1+(−6.61−0.745i)T+(71.1+16.2i)T2 1 + (-6.61 - 0.745i)T + (71.1 + 16.2i)T^{2}
79 1+(11.7−4.10i)T+(61.7−49.2i)T2 1 + (11.7 - 4.10i)T + (61.7 - 49.2i)T^{2}
83 1+(−3.90+0.892i)T+(74.7−36.0i)T2 1 + (-3.90 + 0.892i)T + (74.7 - 36.0i)T^{2}
89 1+(−15.4+1.74i)T+(86.7−19.8i)T2 1 + (-15.4 + 1.74i)T + (86.7 - 19.8i)T^{2}
97 1+(−8.35−5.25i)T+(42.0+87.3i)T2 1 + (-8.35 - 5.25i)T + (42.0 + 87.3i)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

−13.88375019344348553471808378560, −12.82227314395935590512529712286, −12.03624946005720292097796107015, −10.70975833602916864609838204317, −9.216402068519151328783976490396, −8.353531996412876243991698128995, −6.86019321501103044179562503685, −5.96914666080147699709668184739, −4.29655131761876825369040810952, −3.31741976212892889508717250223, 2.06558539952046943745299199582, 3.67899850421372048123041399718, 5.11885682947743076363168292005, 6.54428666702854622551873760565, 7.75128860729808631477646609051, 9.458044461733806357590638990615, 10.30771341140933516075238417059, 11.46057070148825730431387996853, 12.56365294768499641476262908003, 13.12694575174062378158415995422

Graph of the ZZ-function along the critical line