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Properties

Label 2-91-91.9-c1-0-1
Degree 22
Conductor 9191
Sign 0.835−0.549i0.835 - 0.549i
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  + (0.425 + 0.737i)2-s + 0.661·3-s + (0.637 − 1.10i)4-s + (−1.72 + 2.98i)5-s + (0.281 + 0.487i)6-s + (1.82 − 1.91i)7-s + 2.78·8-s − 2.56·9-s − 2.92·10-s − 0.897·11-s + (0.421 − 0.730i)12-s + (−3.07 − 1.88i)13-s + (2.18 + 0.525i)14-s + (−1.13 + 1.97i)15-s + (−0.0891 − 0.154i)16-s + (−0.968 + 1.67i)17-s + ⋯
L(s)  = 1  + (0.300 + 0.521i)2-s + 0.381·3-s + (0.318 − 0.552i)4-s + (−0.769 + 1.33i)5-s + (0.114 + 0.198i)6-s + (0.688 − 0.725i)7-s + 0.985·8-s − 0.854·9-s − 0.926·10-s − 0.270·11-s + (0.121 − 0.210i)12-s + (−0.852 − 0.522i)13-s + (0.585 + 0.140i)14-s + (−0.293 + 0.508i)15-s + (−0.0222 − 0.0386i)16-s + (−0.234 + 0.406i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((0.835−0.549i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((0.835−0.549i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: 0.835−0.549i0.835 - 0.549i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(9,⋅)\chi_{91} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), 0.835−0.549i)(2,\ 91,\ (\ :1/2),\ 0.835 - 0.549i)

Particular Values

L(1)L(1) ≈\approx 1.17060+0.350081i1.17060 + 0.350081i
L(12)L(\frac12) ≈\approx 1.17060+0.350081i1.17060 + 0.350081i
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+(−1.82+1.91i)T 1 + (-1.82 + 1.91i)T
13 1+(3.07+1.88i)T 1 + (3.07 + 1.88i)T
good2 1+(−0.425−0.737i)T+(−1+1.73i)T2 1 + (-0.425 - 0.737i)T + (-1 + 1.73i)T^{2}
3 1−0.661T+3T2 1 - 0.661T + 3T^{2}
5 1+(1.72−2.98i)T+(−2.5−4.33i)T2 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2}
11 1+0.897T+11T2 1 + 0.897T + 11T^{2}
17 1+(0.968−1.67i)T+(−8.5−14.7i)T2 1 + (0.968 - 1.67i)T + (-8.5 - 14.7i)T^{2}
19 1−1.03T+19T2 1 - 1.03T + 19T^{2}
23 1+(2.82+4.89i)T+(−11.5+19.9i)T2 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.917+1.58i)T+(−14.5−25.1i)T2 1 + (-0.917 + 1.58i)T + (-14.5 - 25.1i)T^{2}
31 1+(−4.56−7.91i)T+(−15.5+26.8i)T2 1 + (-4.56 - 7.91i)T + (-15.5 + 26.8i)T^{2}
37 1+(−5.30−9.17i)T+(−18.5+32.0i)T2 1 + (-5.30 - 9.17i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.66+4.61i)T+(−20.5−35.5i)T2 1 + (-2.66 + 4.61i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1.95−3.39i)T+(−21.5+37.2i)T2 1 + (-1.95 - 3.39i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.59−6.22i)T+(−23.5−40.7i)T2 1 + (3.59 - 6.22i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.69−8.12i)T+(−26.5+45.8i)T2 1 + (-4.69 - 8.12i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.255+0.442i)T+(−29.5−51.0i)T2 1 + (-0.255 + 0.442i)T + (-29.5 - 51.0i)T^{2}
61 1−1.43T+61T2 1 - 1.43T + 61T^{2}
67 1+8.44T+67T2 1 + 8.44T + 67T^{2}
71 1+(−1.72−2.98i)T+(−35.5+61.4i)T2 1 + (-1.72 - 2.98i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.45+9.44i)T+(−36.5+63.2i)T2 1 + (5.45 + 9.44i)T + (-36.5 + 63.2i)T^{2}
79 1+(−6.04+10.4i)T+(−39.5−68.4i)T2 1 + (-6.04 + 10.4i)T + (-39.5 - 68.4i)T^{2}
83 1−1.51T+83T2 1 - 1.51T + 83T^{2}
89 1+(6.80+11.7i)T+(−44.5+77.0i)T2 1 + (6.80 + 11.7i)T + (-44.5 + 77.0i)T^{2}
97 1+(0.253+0.438i)T+(−48.5+84.0i)T2 1 + (0.253 + 0.438i)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

−14.52875663996914739134406325524, −13.69426330326917635776142310814, −11.85223180675114560978710229113, −10.83610598699277088580969288796, −10.23783547267479541796982520916, −8.128445280838652079945059846146, −7.37622698324790937842623975584, −6.24132413699861190837959589029, −4.59994424967326469857119611745, −2.80729778271699423842019959130, 2.37414593614358078020455315650, 4.13412758953910039249696777954, 5.31339248127785475489640391337, 7.64071654156067761968538923123, 8.328433463263302207206998795737, 9.387815397312900323903601476874, 11.45232025200309658110770882044, 11.76456721465226781299846671936, 12.71433126338250883585854738774, 13.78353178314597347441636827529

Graph of the ZZ-function along the critical line