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Properties

Label 2-91-7.4-c1-0-3
Degree 22
Conductor 9191
Sign 0.386−0.922i0.386 - 0.922i
Analytic cond. 0.7266380.726638
Root an. cond. 0.8524310.852431
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  + (1.30 + 2.26i)2-s + (1.11 − 1.93i)3-s + (−2.42 + 4.20i)4-s + (−1.11 − 1.93i)5-s + 5.85·6-s + (−2 + 1.73i)7-s − 7.47·8-s + (−1 − 1.73i)9-s + (2.92 − 5.06i)10-s + (1.5 − 2.59i)11-s + (5.42 + 9.39i)12-s − 13-s + (−6.54 − 2.26i)14-s − 5.00·15-s + (−4.92 − 8.53i)16-s + (−0.736 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.925 + 1.60i)2-s + (0.645 − 1.11i)3-s + (−1.21 + 2.10i)4-s + (−0.499 − 0.866i)5-s + 2.38·6-s + (−0.755 + 0.654i)7-s − 2.64·8-s + (−0.333 − 0.577i)9-s + (0.925 − 1.60i)10-s + (0.452 − 0.783i)11-s + (1.56 + 2.71i)12-s − 0.277·13-s + (−1.74 − 0.605i)14-s − 1.29·15-s + (−1.23 − 2.13i)16-s + (−0.178 + 0.309i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(53,⋅)\chi_{91} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), 0.386−0.922i)(2,\ 91,\ (\ :1/2),\ 0.386 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.23124+0.819000i1.23124 + 0.819000i
L(12)L(\frac12) ≈\approx 1.23124+0.819000i1.23124 + 0.819000i
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+(2−1.73i)T 1 + (2 - 1.73i)T
13 1+T 1 + T
good2 1+(−1.30−2.26i)T+(−1+1.73i)T2 1 + (-1.30 - 2.26i)T + (-1 + 1.73i)T^{2}
3 1+(−1.11+1.93i)T+(−1.5−2.59i)T2 1 + (-1.11 + 1.93i)T + (-1.5 - 2.59i)T^{2}
5 1+(1.11+1.93i)T+(−2.5+4.33i)T2 1 + (1.11 + 1.93i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.5+2.59i)T+(−5.5−9.52i)T2 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2}
17 1+(0.736−1.27i)T+(−8.5−14.7i)T2 1 + (0.736 - 1.27i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.5+2.59i)T+(−9.5+16.4i)T2 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2}
23 1+(−4.11−7.13i)T+(−11.5+19.9i)T2 1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2}
29 1−4.47T+29T2 1 - 4.47T + 29T^{2}
31 1+(2.5−4.33i)T+(−15.5−26.8i)T2 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.35+4.07i)T+(−18.5+32.0i)T2 1 + (2.35 + 4.07i)T + (-18.5 + 32.0i)T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1+(−3.73−6.47i)T+(−23.5+40.7i)T2 1 + (-3.73 - 6.47i)T + (-23.5 + 40.7i)T^{2}
53 1+(−3.73+6.47i)T+(−26.5−45.8i)T2 1 + (-3.73 + 6.47i)T + (-26.5 - 45.8i)T^{2}
59 1+(−0.736+1.27i)T+(−29.5−51.0i)T2 1 + (-0.736 + 1.27i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.5+2.59i)T+(−30.5+52.8i)T2 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.5+2.59i)T+(−33.5−58.0i)T2 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2}
71 1+8.94T+71T2 1 + 8.94T + 71T^{2}
73 1+(−1.35+2.34i)T+(−36.5−63.2i)T2 1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.35−2.34i)T+(−39.5+68.4i)T2 1 + (-1.35 - 2.34i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(1.11+1.93i)T+(−44.5+77.0i)T2 1 + (1.11 + 1.93i)T + (-44.5 + 77.0i)T^{2}
97 1−9.41T+97T2 1 - 9.41T + 97T^{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

−14.20626976383485173222061194435, −13.30654486884255639490446195010, −12.76409423261741952593782406923, −11.91510771744855902170491470600, −8.946273652550477929618178825074, −8.458333016419311493913696830101, −7.28707580737416211507804133607, −6.37522657177174766914774324039, −5.05436121462349775984850677814, −3.35327604878179764201177913968, 2.82081953332938905818375982573, 3.77734419038093641113834944118, 4.64406321105409139521068003200, 6.77337655772861857255230814779, 9.025745998523202722389118705428, 10.19511879307132675294626325601, 10.41507271308888056682771179061, 11.71423185081363697482551012281, 12.75344849804855770208887521365, 13.89616788215796578725131254382

Graph of the ZZ-function along the critical line