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Properties

Label 2-91-13.3-c1-0-0
Degree 22
Conductor 9191
Sign 0.859+0.511i0.859 + 0.511i
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 + (1.30 + 2.26i)3-s + (−2.42 + 4.20i)4-s + 2.61·5-s + (3.42 − 5.93i)6-s + (−0.5 + 0.866i)7-s + 7.47·8-s + (−1.92 + 3.33i)9-s + (−3.42 − 5.93i)10-s + (−0.927 − 1.60i)11-s − 12.7·12-s + (−2.5 − 2.59i)13-s + 2.61·14-s + (3.42 + 5.93i)15-s + (−4.92 − 8.53i)16-s + (0.736 − 1.27i)17-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)2-s + (0.755 + 1.30i)3-s + (−1.21 + 2.10i)4-s + 1.17·5-s + (1.39 − 2.42i)6-s + (−0.188 + 0.327i)7-s + 2.64·8-s + (−0.642 + 1.11i)9-s + (−1.08 − 1.87i)10-s + (−0.279 − 0.484i)11-s − 3.66·12-s + (−0.693 − 0.720i)13-s + 0.699·14-s + (0.884 + 1.53i)15-s + (−1.23 − 2.13i)16-s + (0.178 − 0.309i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((0.859+0.511i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((0.859+0.511i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: 0.859+0.511i0.859 + 0.511i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(29,⋅)\chi_{91} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), 0.859+0.511i)(2,\ 91,\ (\ :1/2),\ 0.859 + 0.511i)

Particular Values

L(1)L(1) ≈\approx 0.801490−0.220276i0.801490 - 0.220276i
L(12)L(\frac12) ≈\approx 0.801490−0.220276i0.801490 - 0.220276i
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
13 1+(2.5+2.59i)T 1 + (2.5 + 2.59i)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+(−1.30−2.26i)T+(−1.5+2.59i)T2 1 + (-1.30 - 2.26i)T + (-1.5 + 2.59i)T^{2}
5 1−2.61T+5T2 1 - 2.61T + 5T^{2}
11 1+(0.927+1.60i)T+(−5.5+9.52i)T2 1 + (0.927 + 1.60i)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+(−0.927+1.60i)T+(−9.5−16.4i)T2 1 + (-0.927 + 1.60i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.23−3.87i)T+(−11.5+19.9i)T2 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.54+6.14i)T+(−14.5+25.1i)T2 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2}
31 1+4.70T+31T2 1 + 4.70T + 31T^{2}
37 1+(2+3.46i)T+(−18.5+32.0i)T2 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.381+0.661i)T+(−20.5+35.5i)T2 1 + (0.381 + 0.661i)T + (-20.5 + 35.5i)T^{2}
43 1+(6.28−10.8i)T+(−21.5−37.2i)T2 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2}
47 1+2.23T+47T2 1 + 2.23T + 47T^{2}
53 1−3.76T+53T2 1 - 3.76T + 53T^{2}
59 1+(−1.11+1.93i)T+(−29.5−51.0i)T2 1 + (-1.11 + 1.93i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3+5.19i)T+(−30.5−52.8i)T2 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.35−11.0i)T+(−33.5+58.0i)T2 1 + (-6.35 - 11.0i)T + (-33.5 + 58.0i)T^{2}
71 1+(−7.09+12.2i)T+(−35.5−61.4i)T2 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−6.70T+83T2 1 - 6.70T + 83T^{2}
89 1+(2.45+4.25i)T+(−44.5+77.0i)T2 1 + (2.45 + 4.25i)T + (-44.5 + 77.0i)T^{2}
97 1+(9.42−16.3i)T+(−48.5−84.0i)T2 1 + (9.42 - 16.3i)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

−13.69585316563607396005583691059, −12.88742228761098839182673088396, −11.41799207551908262378884258618, −10.40739824166229282417700423445, −9.607390863362311771317061218147, −9.265014826457146716128406306395, −8.006658353264053116592906854008, −5.21395122041646884883505643151, −3.47108356443198913441640188356, −2.42080627102110695534675623945, 1.80214490826245528977902820077, 5.36404778182236042436944321575, 6.70797965088300389274497248723, 7.20714144755062709813776420676, 8.402632954584516310066139809028, 9.348664020825086494977289568415, 10.27514006967594240862702061569, 12.61119931919059129964950553941, 13.60944668620195947049057451698, 14.27003126846218231628705924284

Graph of the ZZ-function along the critical line