↑

Properties

Label 2-3762-33.32-c1-0-22
Degree 22
Conductor 37623762
Sign 0.0163−0.999i0.0163 - 0.999i
Analytic cond. 30.039730.0397
Root an. cond. 5.480855.48085
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  − 2-s + 4-s + 1.60i·5-s + 1.65i·7-s − 8-s − 1.60i·10-s + (1.87 + 2.73i)11-s + 1.23i·13-s − 1.65i·14-s + 16-s − 1.74·17-s + i·19-s + 1.60i·20-s + (−1.87 − 2.73i)22-s − 5.99i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.715i·5-s + 0.625i·7-s − 0.353·8-s − 0.506i·10-s + (0.563 + 0.825i)11-s + 0.343i·13-s − 0.442i·14-s + 0.250·16-s − 0.423·17-s + 0.229i·19-s + 0.357i·20-s + (−0.398 − 0.583i)22-s − 1.24i·23-s + ⋯

Functional equation

Λ(s)=(3762s/2 ΓC(s) L(s)=((0.0163−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0163 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3762s/2 ΓC(s+1/2) L(s)=((0.0163−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0163 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37623762    =    2⋅32⋅11⋅192 \cdot 3^{2} \cdot 11 \cdot 19
Sign: 0.0163−0.999i0.0163 - 0.999i
Analytic conductor: 30.039730.0397
Root analytic conductor: 5.480855.48085
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3762(989,⋅)\chi_{3762} (989, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3762, ( :1/2), 0.0163−0.999i)(2,\ 3762,\ (\ :1/2),\ 0.0163 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.4029858401.402985840
L(12)L(\frac12) ≈\approx 1.4029858401.402985840
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+T 1 + T
3 1 1
11 1+(−1.87−2.73i)T 1 + (-1.87 - 2.73i)T
19 1−iT 1 - iT
good5 1−1.60iT−5T2 1 - 1.60iT - 5T^{2}
7 1−1.65iT−7T2 1 - 1.65iT - 7T^{2}
13 1−1.23iT−13T2 1 - 1.23iT - 13T^{2}
17 1+1.74T+17T2 1 + 1.74T + 17T^{2}
23 1+5.99iT−23T2 1 + 5.99iT - 23T^{2}
29 1−7.25T+29T2 1 - 7.25T + 29T^{2}
31 1−10.5T+31T2 1 - 10.5T + 31T^{2}
37 1−7.98T+37T2 1 - 7.98T + 37T^{2}
41 1+8.60T+41T2 1 + 8.60T + 41T^{2}
43 1+5.30iT−43T2 1 + 5.30iT - 43T^{2}
47 1−2.24iT−47T2 1 - 2.24iT - 47T^{2}
53 1−4.01iT−53T2 1 - 4.01iT - 53T^{2}
59 1−10.9iT−59T2 1 - 10.9iT - 59T^{2}
61 1+0.355iT−61T2 1 + 0.355iT - 61T^{2}
67 1−9.96T+67T2 1 - 9.96T + 67T^{2}
71 1−8.44iT−71T2 1 - 8.44iT - 71T^{2}
73 1−1.24iT−73T2 1 - 1.24iT - 73T^{2}
79 1+1.72iT−79T2 1 + 1.72iT - 79T^{2}
83 1−8.04T+83T2 1 - 8.04T + 83T^{2}
89 1+4.10iT−89T2 1 + 4.10iT - 89T^{2}
97 1+3.14T+97T2 1 + 3.14T + 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

−8.620212270785293130436412624412, −8.167195716607342972661773368454, −7.05610566117266640061331497629, −6.65905789420423989706152859853, −6.05899689156203573665916065628, −4.84483790529076287705667215532, −4.12415036203119460946580307309, −2.78666272506766606916506376319, −2.37211655044438837294615234623, −1.08153439038145249076999474908, 0.65925847946975348974680297353, 1.30704780455167519631349359702, 2.68958510030668742055065821860, 3.58440010608028372191687678253, 4.55653083859008814768608233450, 5.30728906562932161581042534220, 6.40174053736171140916674536068, 6.75443112680197751282500271427, 7.954002652594793023036410874016, 8.230634864030640758955405388889

Graph of the ZZ-function along the critical line