↑

Properties

Label 2-460-92.91-c1-0-47
Degree 22
Conductor 460460
Sign 0.182−0.983i0.182 - 0.983i
Analytic cond. 3.673113.67311
Root an. cond. 1.916531.91653
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  + (−0.151 − 1.40i)2-s − 2.99i·3-s + (−1.95 + 0.425i)4-s + i·5-s + (−4.21 + 0.453i)6-s − 3.98·7-s + (0.893 + 2.68i)8-s − 5.98·9-s + (1.40 − 0.151i)10-s + 5.79·11-s + (1.27 + 5.85i)12-s − 5.59·13-s + (0.602 + 5.59i)14-s + 2.99·15-s + (3.63 − 1.66i)16-s + 1.88i·17-s + ⋯
L(s)  = 1  + (−0.106 − 0.994i)2-s − 1.73i·3-s + (−0.977 + 0.212i)4-s + 0.447i·5-s + (−1.72 + 0.185i)6-s − 1.50·7-s + (0.315 + 0.948i)8-s − 1.99·9-s + (0.444 − 0.0478i)10-s + 1.74·11-s + (0.368 + 1.69i)12-s − 1.55·13-s + (0.160 + 1.49i)14-s + 0.774·15-s + (0.909 − 0.415i)16-s + 0.456i·17-s + ⋯

Functional equation

Λ(s)=(460s/2 ΓC(s) L(s)=((0.182−0.983i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(460s/2 ΓC(s+1/2) L(s)=((0.182−0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 460460    =    22⋅5⋅232^{2} \cdot 5 \cdot 23
Sign: 0.182−0.983i0.182 - 0.983i
Analytic conductor: 3.673113.67311
Root analytic conductor: 1.916531.91653
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ460(91,⋅)\chi_{460} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 460, ( :1/2), 0.182−0.983i)(2,\ 460,\ (\ :1/2),\ 0.182 - 0.983i)

Particular Values

L(1)L(1) ≈\approx 0.235852+0.196028i0.235852 + 0.196028i
L(12)L(\frac12) ≈\approx 0.235852+0.196028i0.235852 + 0.196028i
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+(0.151+1.40i)T 1 + (0.151 + 1.40i)T
5 1−iT 1 - iT
23 1+(4.42+1.85i)T 1 + (4.42 + 1.85i)T
good3 1+2.99iT−3T2 1 + 2.99iT - 3T^{2}
7 1+3.98T+7T2 1 + 3.98T + 7T^{2}
11 1−5.79T+11T2 1 - 5.79T + 11T^{2}
13 1+5.59T+13T2 1 + 5.59T + 13T^{2}
17 1−1.88iT−17T2 1 - 1.88iT - 17T^{2}
19 1+3.03T+19T2 1 + 3.03T + 19T^{2}
29 1−1.45T+29T2 1 - 1.45T + 29T^{2}
31 1+4.43iT−31T2 1 + 4.43iT - 31T^{2}
37 1+4.01iT−37T2 1 + 4.01iT - 37T^{2}
41 1+2.68T+41T2 1 + 2.68T + 41T^{2}
43 1+4.78T+43T2 1 + 4.78T + 43T^{2}
47 1+6.45iT−47T2 1 + 6.45iT - 47T^{2}
53 1−8.62iT−53T2 1 - 8.62iT - 53T^{2}
59 1+1.55iT−59T2 1 + 1.55iT - 59T^{2}
61 1−6.06iT−61T2 1 - 6.06iT - 61T^{2}
67 1+7.88T+67T2 1 + 7.88T + 67T^{2}
71 1+5.26iT−71T2 1 + 5.26iT - 71T^{2}
73 1−2.20T+73T2 1 - 2.20T + 73T^{2}
79 1+13.1T+79T2 1 + 13.1T + 79T^{2}
83 1−8.56T+83T2 1 - 8.56T + 83T^{2}
89 1+8.41iT−89T2 1 + 8.41iT - 89T^{2}
97 1+2.87iT−97T2 1 + 2.87iT - 97T^{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

−10.35482405790559275446168623264, −9.532772467820616936686753714020, −8.667149571878119689777037207688, −7.49285490330461728754238410171, −6.67011730253644704684763287600, −5.98404387249697065529017982145, −4.04188197041880277124225252415, −2.84512112828309356721052076148, −1.86716496974723538066563430294, −0.19012666653588818652717362596, 3.34110019554646492895408256571, 4.21188387190734249642296689289, 5.01290255646787884011718975729, 6.14199083626223679851200041030, 6.93471034886271477760671154014, 8.458340928756191364566344478084, 9.342951710753905099087750395439, 9.652819260628648970225242499842, 10.25882349590092463835416770233, 11.77955863727769696459167501464

Graph of the ZZ-function along the critical line