Properties

Label 2-91-13.10-c1-0-3
Degree $2$
Conductor $91$
Sign $0.992 + 0.124i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 − 0.638i)2-s + (0.583 + 1.01i)3-s + (−0.185 + 0.320i)4-s − 1.81i·5-s + (1.29 + 0.745i)6-s + (−0.866 − 0.5i)7-s + 3.02i·8-s + (0.817 − 1.41i)9-s + (−1.15 − 2.00i)10-s + (−2.40 + 1.38i)11-s − 0.432·12-s + (−3.58 − 0.402i)13-s − 1.27·14-s + (1.83 − 1.05i)15-s + (1.56 + 2.70i)16-s + (1.37 − 2.37i)17-s + ⋯
L(s)  = 1  + (0.781 − 0.451i)2-s + (0.337 + 0.583i)3-s + (−0.0925 + 0.160i)4-s − 0.811i·5-s + (0.527 + 0.304i)6-s + (−0.327 − 0.188i)7-s + 1.06i·8-s + (0.272 − 0.472i)9-s + (−0.366 − 0.634i)10-s + (−0.725 + 0.418i)11-s − 0.124·12-s + (−0.993 − 0.111i)13-s − 0.341·14-s + (0.473 − 0.273i)15-s + (0.390 + 0.675i)16-s + (0.332 − 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38410 - 0.0863373i\)
\(L(\frac12)\) \(\approx\) \(1.38410 - 0.0863373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (3.58 + 0.402i)T \)
good2 \( 1 + (-1.10 + 0.638i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.583 - 1.01i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.81iT - 5T^{2} \)
11 \( 1 + (2.40 - 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.37 + 2.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.08 + 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.49 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + (-1.50 + 0.871i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.51 + 3.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.55 + 7.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-2.66 - 1.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.540 - 0.936i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.34 + 2.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.35 - 1.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.67iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 7.97iT - 83T^{2} \)
89 \( 1 + (13.9 - 8.03i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.3 + 7.11i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.94120858849044619493196893023, −12.67006774427715612824133196247, −12.52688054481867431628106739031, −10.96351849642793593646094995506, −9.652912437343926667110635508283, −8.759136980679703780296429751654, −7.30793886121906200547190537351, −5.19034205879090235256616099233, −4.35016630847608067248626968969, −2.90028788394242707442921200773, 2.69613756933845690814656343808, 4.56441073225190743749789192384, 6.06402511012631058446663523541, 7.01556215633768827802651579990, 8.212829479247013350517752280430, 9.936512660969266685594777179247, 10.77742982400045133063075585048, 12.64329798658626400512978795346, 13.03388282931873104906524083169, 14.36366394300318477523139008864

Graph of the $Z$-function along the critical line