↑

Properties

Label 2-150-15.14-c2-0-5
Degree 22
Conductor 150150
Sign 0.541−0.840i0.541 - 0.840i
Analytic cond. 4.087204.08720
Root an. cond. 2.021682.02168
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (1.52 + 2.58i)3-s + 2.00·4-s + (2.16 + 3.65i)6-s + 7.48i·7-s + 2.82·8-s + (−4.32 + 7.89i)9-s − 8.48i·11-s + (3.05 + 5.16i)12-s − 10i·13-s + 10.5i·14-s + 4.00·16-s + 30.3·17-s + (−6.11 + 11.1i)18-s − 26.9·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.509 + 0.860i)3-s + 0.500·4-s + (0.360 + 0.608i)6-s + 1.06i·7-s + 0.353·8-s + (−0.480 + 0.876i)9-s − 0.771i·11-s + (0.254 + 0.430i)12-s − 0.769i·13-s + 0.756i·14-s + 0.250·16-s + 1.78·17-s + (−0.339 + 0.620i)18-s − 1.41·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.541−0.840i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+1) L(s)=((0.541−0.840i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.541−0.840i0.541 - 0.840i
Analytic conductor: 4.087204.08720
Root analytic conductor: 2.021682.02168
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ150(149,⋅)\chi_{150} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :1), 0.541−0.840i)(2,\ 150,\ (\ :1),\ 0.541 - 0.840i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.16121+1.17847i2.16121 + 1.17847i
L(12)L(\frac12) ≈\approx 2.16121+1.17847i2.16121 + 1.17847i
L(2)L(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 1−1.41T 1 - 1.41T
3 1+(−1.52−2.58i)T 1 + (-1.52 - 2.58i)T
5 1 1
good7 1−7.48iT−49T2 1 - 7.48iT - 49T^{2}
11 1+8.48iT−121T2 1 + 8.48iT - 121T^{2}
13 1+10iT−169T2 1 + 10iT - 169T^{2}
17 1−30.3T+289T2 1 - 30.3T + 289T^{2}
19 1+26.9T+361T2 1 + 26.9T + 361T^{2}
23 1−9.17T+529T2 1 - 9.17T + 529T^{2}
29 1+26.8iT−841T2 1 + 26.8iT - 841T^{2}
31 1−8T+961T2 1 - 8T + 961T^{2}
37 1+15.9iT−1.36e3T2 1 + 15.9iT - 1.36e3T^{2}
41 1+47.3iT−1.68e3T2 1 + 47.3iT - 1.68e3T^{2}
43 1+14.4iT−1.84e3T2 1 + 14.4iT - 1.84e3T^{2}
47 1+45.8T+2.20e3T2 1 + 45.8T + 2.20e3T^{2}
53 1+30.3T+2.80e3T2 1 + 30.3T + 2.80e3T^{2}
59 1+24.0iT−3.48e3T2 1 + 24.0iT - 3.48e3T^{2}
61 1+53.9T+3.72e3T2 1 + 53.9T + 3.72e3T^{2}
67 1−110.iT−4.48e3T2 1 - 110. iT - 4.48e3T^{2}
71 1+15.5iT−5.04e3T2 1 + 15.5iT - 5.04e3T^{2}
73 1−87.9iT−5.32e3T2 1 - 87.9iT - 5.32e3T^{2}
79 1−46.9T+6.24e3T2 1 - 46.9T + 6.24e3T^{2}
83 1−26.1T+6.88e3T2 1 - 26.1T + 6.88e3T^{2}
89 1−60.7iT−7.92e3T2 1 - 60.7iT - 7.92e3T^{2}
97 1+36.0iT−9.40e3T2 1 + 36.0iT - 9.40e3T^{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

−12.98314935262949686336331196059, −12.03560688224170915022710444779, −10.92652949290726808765235874023, −9.993352591803428770687886484087, −8.749427492223122272097870633545, −7.896578276129167112954996531661, −5.98664784951455197102931214773, −5.21076881641888786069541884885, −3.69558346587960620401120730404, −2.58926661467084293340603813935, 1.55235360461407244243748556206, 3.28628954233851860633489864972, 4.59682451486433025647897038491, 6.32184853307430497579382756250, 7.18726871974569952408570515948, 8.089423692445458428968936679018, 9.588874636595068378878493523961, 10.74665530356542888354935279185, 12.00337832099193586772924188548, 12.72930667842318176411271279812

Graph of the ZZ-function along the critical line