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Properties

Label 2-133-133.102-c1-0-6
Degree 22
Conductor 133133
Sign 0.962+0.271i0.962 + 0.271i
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  + (−1.16 + 2.01i)2-s − 1.27·3-s + (−1.69 − 2.93i)4-s + (2.05 − 3.55i)5-s + (1.48 − 2.56i)6-s + (−0.208 − 2.63i)7-s + 3.23·8-s − 1.36·9-s + (4.76 + 8.25i)10-s + (−1.36 + 2.36i)11-s + (2.16 + 3.75i)12-s + (2.05 − 3.55i)13-s + (5.54 + 2.64i)14-s + (−2.61 + 4.53i)15-s + (−0.360 + 0.624i)16-s + 1.24·17-s + ⋯
L(s)  = 1  + (−0.820 + 1.42i)2-s − 0.737·3-s + (−0.847 − 1.46i)4-s + (0.917 − 1.58i)5-s + (0.605 − 1.04i)6-s + (−0.0789 − 0.996i)7-s + 1.14·8-s − 0.456·9-s + (1.50 + 2.60i)10-s + (−0.412 + 0.713i)11-s + (0.625 + 1.08i)12-s + (0.569 − 0.986i)13-s + (1.48 + 0.706i)14-s + (−0.676 + 1.17i)15-s + (−0.0901 + 0.156i)16-s + 0.302·17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((0.962+0.271i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((0.962+0.271i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: 0.962+0.271i0.962 + 0.271i
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), 0.962+0.271i)(2,\ 133,\ (\ :1/2),\ 0.962 + 0.271i)

Particular Values

L(1)L(1) ≈\approx 0.529041−0.0732882i0.529041 - 0.0732882i
L(12)L(\frac12) ≈\approx 0.529041−0.0732882i0.529041 - 0.0732882i
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+(0.208+2.63i)T 1 + (0.208 + 2.63i)T
19 1+(4.34+0.339i)T 1 + (4.34 + 0.339i)T
good2 1+(1.16−2.01i)T+(−1−1.73i)T2 1 + (1.16 - 2.01i)T + (-1 - 1.73i)T^{2}
3 1+1.27T+3T2 1 + 1.27T + 3T^{2}
5 1+(−2.05+3.55i)T+(−2.5−4.33i)T2 1 + (-2.05 + 3.55i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.36−2.36i)T+(−5.5−9.52i)T2 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.05+3.55i)T+(−6.5−11.2i)T2 1 + (-2.05 + 3.55i)T + (-6.5 - 11.2i)T^{2}
17 1−1.24T+17T2 1 - 1.24T + 17T^{2}
23 1−5.63T+23T2 1 - 5.63T + 23T^{2}
29 1+(−0.386+0.669i)T+(−14.5−25.1i)T2 1 + (-0.386 + 0.669i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.21−2.11i)T+(−15.5−26.8i)T2 1 + (1.21 - 2.11i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.32+2.28i)T+(−18.5+32.0i)T2 1 + (1.32 + 2.28i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.41−4.17i)T+(−20.5+35.5i)T2 1 + (-2.41 - 4.17i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.250+0.434i)T+(−21.5+37.2i)T2 1 + (0.250 + 0.434i)T + (-21.5 + 37.2i)T^{2}
47 1−0.00235T+47T2 1 - 0.00235T + 47T^{2}
53 1+(−0.280−0.485i)T+(−26.5+45.8i)T2 1 + (-0.280 - 0.485i)T + (-26.5 + 45.8i)T^{2}
59 1+2.61T+59T2 1 + 2.61T + 59T^{2}
61 1−11.7T+61T2 1 - 11.7T + 61T^{2}
67 1+(4.70+8.14i)T+(−33.5+58.0i)T2 1 + (4.70 + 8.14i)T + (-33.5 + 58.0i)T^{2}
71 1+(−5.28−9.14i)T+(−35.5+61.4i)T2 1 + (-5.28 - 9.14i)T + (-35.5 + 61.4i)T^{2}
73 1−11.2T+73T2 1 - 11.2T + 73T^{2}
79 1+(0.675−1.16i)T+(−39.5−68.4i)T2 1 + (0.675 - 1.16i)T + (-39.5 - 68.4i)T^{2}
83 1−6.83T+83T2 1 - 6.83T + 83T^{2}
89 1+7.59T+89T2 1 + 7.59T + 89T^{2}
97 1+(−1.72−2.99i)T+(−48.5+84.0i)T2 1 + (-1.72 - 2.99i)T + (-48.5 + 84.0i)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

−13.24294988989198606206992908160, −12.53843400095987329680641795842, −10.75121288514408208183277255650, −9.836199898554764562327304919136, −8.804204364972950380621275941300, −7.970444161792871124326352868422, −6.60641119191435892258125879673, −5.58213036173515321544890304138, −4.85113713332117113547854454375, −0.802541341592220561337579294404, 2.21308869103481245661817628151, 3.21255923565107538142613056924, 5.70952394450989302493954391839, 6.59288062740105341781257303069, 8.537091325016231210903393437425, 9.442588985094819343456381296663, 10.59875838895512652372023038635, 11.06882070731908241986177455782, 11.75308655083556624102569435698, 12.97103781302439113454228674620

Graph of the ZZ-function along the critical line