↑

Properties

Label 2-133-133.102-c1-0-5
Degree 22
Conductor 133133
Sign −0.0416−0.999i-0.0416 - 0.999i
Analytic cond. 1.062011.06201
Root an. cond. 1.030531.03053
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  + (−1.14 + 1.97i)2-s + 2.46·3-s + (−1.60 − 2.77i)4-s + (0.527 − 0.913i)5-s + (−2.81 + 4.86i)6-s + (1.23 + 2.33i)7-s + 2.74·8-s + 3.07·9-s + (1.20 + 2.08i)10-s + (0.0883 − 0.152i)11-s + (−3.94 − 6.83i)12-s + (−0.270 + 0.468i)13-s + (−6.03 − 0.226i)14-s + (1.29 − 2.25i)15-s + (0.0736 − 0.127i)16-s − 7.92·17-s + ⋯
L(s)  = 1  + (−0.806 + 1.39i)2-s + 1.42·3-s + (−0.800 − 1.38i)4-s + (0.235 − 0.408i)5-s + (−1.14 + 1.98i)6-s + (0.467 + 0.884i)7-s + 0.970·8-s + 1.02·9-s + (0.380 + 0.658i)10-s + (0.0266 − 0.0461i)11-s + (−1.13 − 1.97i)12-s + (−0.0750 + 0.130i)13-s + (−1.61 − 0.0606i)14-s + (0.335 − 0.581i)15-s + (0.0184 − 0.0319i)16-s − 1.92·17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((−0.0416−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0416 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((−0.0416−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0416 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: −0.0416−0.999i-0.0416 - 0.999i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ133(102,⋅)\chi_{133} (102, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), −0.0416−0.999i)(2,\ 133,\ (\ :1/2),\ -0.0416 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.761646+0.794051i0.761646 + 0.794051i
L(12)L(\frac12) ≈\approx 0.761646+0.794051i0.761646 + 0.794051i
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)
bad7 1+(−1.23−2.33i)T 1 + (-1.23 - 2.33i)T
19 1+(−0.729+4.29i)T 1 + (-0.729 + 4.29i)T
good2 1+(1.14−1.97i)T+(−1−1.73i)T2 1 + (1.14 - 1.97i)T + (-1 - 1.73i)T^{2}
3 1−2.46T+3T2 1 - 2.46T + 3T^{2}
5 1+(−0.527+0.913i)T+(−2.5−4.33i)T2 1 + (-0.527 + 0.913i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.0883+0.152i)T+(−5.5−9.52i)T2 1 + (-0.0883 + 0.152i)T + (-5.5 - 9.52i)T^{2}
13 1+(0.270−0.468i)T+(−6.5−11.2i)T2 1 + (0.270 - 0.468i)T + (-6.5 - 11.2i)T^{2}
17 1+7.92T+17T2 1 + 7.92T + 17T^{2}
23 1+1.47T+23T2 1 + 1.47T + 23T^{2}
29 1+(−2.19+3.80i)T+(−14.5−25.1i)T2 1 + (-2.19 + 3.80i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.58+6.21i)T+(−15.5−26.8i)T2 1 + (-3.58 + 6.21i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.49+7.79i)T+(−18.5+32.0i)T2 1 + (4.49 + 7.79i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.27−2.21i)T+(−20.5+35.5i)T2 1 + (-1.27 - 2.21i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.11−8.85i)T+(−21.5+37.2i)T2 1 + (-5.11 - 8.85i)T + (-21.5 + 37.2i)T^{2}
47 1+4.48T+47T2 1 + 4.48T + 47T^{2}
53 1+(−0.716−1.24i)T+(−26.5+45.8i)T2 1 + (-0.716 - 1.24i)T + (-26.5 + 45.8i)T^{2}
59 1+0.933T+59T2 1 + 0.933T + 59T^{2}
61 1−10.5T+61T2 1 - 10.5T + 61T^{2}
67 1+(1.65+2.86i)T+(−33.5+58.0i)T2 1 + (1.65 + 2.86i)T + (-33.5 + 58.0i)T^{2}
71 1+(5.22+9.05i)T+(−35.5+61.4i)T2 1 + (5.22 + 9.05i)T + (-35.5 + 61.4i)T^{2}
73 1+8.45T+73T2 1 + 8.45T + 73T^{2}
79 1+(4.94−8.57i)T+(−39.5−68.4i)T2 1 + (4.94 - 8.57i)T + (-39.5 - 68.4i)T^{2}
83 1−0.0234T+83T2 1 - 0.0234T + 83T^{2}
89 1−3.93T+89T2 1 - 3.93T + 89T^{2}
97 1+(−6.55−11.3i)T+(−48.5+84.0i)T2 1 + (-6.55 - 11.3i)T + (-48.5 + 84.0i)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

−13.87797197314572256003515531674, −13.01018589200595821210767957567, −11.35343731858558977806061239767, −9.553833385124880003220297017029, −8.978866326683544737912338572353, −8.408480319777834748125570240318, −7.40237489887255213941655152945, −6.13950207545511351998702446384, −4.68782392012014183685411774876, −2.41176477324404986242070148205, 1.83559285462658029284416800332, 3.02693164854195143225574107862, 4.21842534287425453010921451939, 6.94159249693544758731272238639, 8.289671375156954825649406749629, 8.821228926028793930631266085109, 10.09447114985579232749928173943, 10.58637155046149980851794116778, 11.77968015148203796724399842289, 13.02102636575277518094462837593

Graph of the ZZ-function along the critical line