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Properties

Label 2-133-133.100-c1-0-5
Degree 22
Conductor 133133
Sign 0.211−0.977i0.211 - 0.977i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.358 + 2.03i)2-s + (1.19 − 0.435i)3-s + (−2.13 − 0.776i)4-s + (2.57 − 0.935i)5-s + (0.457 + 2.59i)6-s + (2.17 − 1.50i)7-s + (0.278 − 0.481i)8-s + (−1.05 + 0.885i)9-s + (0.981 + 5.56i)10-s − 5.30·11-s − 2.89·12-s + (−0.214 − 1.21i)13-s + (2.29 + 4.96i)14-s + (2.67 − 2.24i)15-s + (−2.59 − 2.17i)16-s + (−4.42 − 3.71i)17-s + ⋯
L(s)  = 1  + (−0.253 + 1.43i)2-s + (0.691 − 0.251i)3-s + (−1.06 − 0.388i)4-s + (1.14 − 0.418i)5-s + (0.186 + 1.05i)6-s + (0.821 − 0.570i)7-s + (0.0983 − 0.170i)8-s + (−0.351 + 0.295i)9-s + (0.310 + 1.76i)10-s − 1.60·11-s − 0.834·12-s + (−0.0595 − 0.337i)13-s + (0.612 + 1.32i)14-s + (0.689 − 0.578i)15-s + (−0.649 − 0.544i)16-s + (−1.07 − 0.901i)17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((0.211−0.977i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((0.211−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: 0.211−0.977i0.211 - 0.977i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ133(100,⋅)\chi_{133} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), 0.211−0.977i)(2,\ 133,\ (\ :1/2),\ 0.211 - 0.977i)

Particular Values

L(1)L(1) ≈\approx 0.988398+0.797781i0.988398 + 0.797781i
L(12)L(\frac12) ≈\approx 0.988398+0.797781i0.988398 + 0.797781i
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+(−2.17+1.50i)T 1 + (-2.17 + 1.50i)T
19 1+(−1.98−3.87i)T 1 + (-1.98 - 3.87i)T
good2 1+(0.358−2.03i)T+(−1.87−0.684i)T2 1 + (0.358 - 2.03i)T + (-1.87 - 0.684i)T^{2}
3 1+(−1.19+0.435i)T+(2.29−1.92i)T2 1 + (-1.19 + 0.435i)T + (2.29 - 1.92i)T^{2}
5 1+(−2.57+0.935i)T+(3.83−3.21i)T2 1 + (-2.57 + 0.935i)T + (3.83 - 3.21i)T^{2}
11 1+5.30T+11T2 1 + 5.30T + 11T^{2}
13 1+(0.214+1.21i)T+(−12.2+4.44i)T2 1 + (0.214 + 1.21i)T + (-12.2 + 4.44i)T^{2}
17 1+(4.42+3.71i)T+(2.95+16.7i)T2 1 + (4.42 + 3.71i)T + (2.95 + 16.7i)T^{2}
23 1+(−0.994−5.64i)T+(−21.6+7.86i)T2 1 + (-0.994 - 5.64i)T + (-21.6 + 7.86i)T^{2}
29 1+(−1.96−0.714i)T+(22.2+18.6i)T2 1 + (-1.96 - 0.714i)T + (22.2 + 18.6i)T^{2}
31 1+(−2.21+3.83i)T+(−15.5−26.8i)T2 1 + (-2.21 + 3.83i)T + (-15.5 - 26.8i)T^{2}
37 1+(−3.92+6.80i)T+(−18.5−32.0i)T2 1 + (-3.92 + 6.80i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.533−3.02i)T+(−38.5−14.0i)T2 1 + (0.533 - 3.02i)T + (-38.5 - 14.0i)T^{2}
43 1+(4.27+3.58i)T+(7.46+42.3i)T2 1 + (4.27 + 3.58i)T + (7.46 + 42.3i)T^{2}
47 1+(−0.527+0.442i)T+(8.16−46.2i)T2 1 + (-0.527 + 0.442i)T + (8.16 - 46.2i)T^{2}
53 1+(−6.70−2.43i)T+(40.6+34.0i)T2 1 + (-6.70 - 2.43i)T + (40.6 + 34.0i)T^{2}
59 1+(0.436+0.366i)T+(10.2+58.1i)T2 1 + (0.436 + 0.366i)T + (10.2 + 58.1i)T^{2}
61 1+(0.920+5.22i)T+(−57.3+20.8i)T2 1 + (0.920 + 5.22i)T + (-57.3 + 20.8i)T^{2}
67 1+(−1.98−11.2i)T+(−62.9+22.9i)T2 1 + (-1.98 - 11.2i)T + (-62.9 + 22.9i)T^{2}
71 1+(−1.10−0.923i)T+(12.3+69.9i)T2 1 + (-1.10 - 0.923i)T + (12.3 + 69.9i)T^{2}
73 1+(5.59−2.03i)T+(55.9−46.9i)T2 1 + (5.59 - 2.03i)T + (55.9 - 46.9i)T^{2}
79 1+(1.30+1.09i)T+(13.7+77.7i)T2 1 + (1.30 + 1.09i)T + (13.7 + 77.7i)T^{2}
83 1+(−2.46−4.26i)T+(−41.5+71.8i)T2 1 + (-2.46 - 4.26i)T + (-41.5 + 71.8i)T^{2}
89 1+(−3.60−1.31i)T+(68.1+57.2i)T2 1 + (-3.60 - 1.31i)T + (68.1 + 57.2i)T^{2}
97 1+(−15.3+5.57i)T+(74.3−62.3i)T2 1 + (-15.3 + 5.57i)T + (74.3 - 62.3i)T^{2}
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   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

−13.59979228592944637183688331285, −13.29568925353566692997229146759, −11.31307282008273825874468359204, −10.00068774701502946706806534267, −8.897195466495376485244764399860, −7.959295570372617412651631796629, −7.35114175848249063384931607639, −5.68395930430540903131188843447, −5.05384418578623338226522362580, −2.36624254844298824137894510941, 2.20434127801101362294961752223, 2.82732694132616517160864287381, 4.75966887373194352546907322391, 6.33046931188817725147556116272, 8.341898501992835609112638793099, 9.073167675843154135735394464607, 10.15255576755542064902108473859, 10.80418878216845790599696266339, 11.83656364266971606076353926111, 13.06904215944073282220427518433

Graph of the ZZ-function along the critical line