Properties

Label 2-91-91.51-c1-0-5
Degree $2$
Conductor $91$
Sign $0.991 - 0.131i$
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  + (1.97 + 1.14i)2-s + (−1.57 − 2.72i)3-s + (1.61 + 2.78i)4-s + (1.84 + 1.06i)5-s − 7.19i·6-s + (−2.62 + 0.331i)7-s + 2.78i·8-s + (−3.46 + 5.99i)9-s + (2.42 + 4.20i)10-s + (0.267 − 0.154i)11-s + (5.07 − 8.78i)12-s + (−3.22 + 1.62i)13-s + (−5.57 − 2.34i)14-s − 6.69i·15-s + (0.0349 − 0.0605i)16-s + (−0.887 − 1.53i)17-s + ⋯
L(s)  = 1  + (1.39 + 0.807i)2-s + (−0.909 − 1.57i)3-s + (0.805 + 1.39i)4-s + (0.823 + 0.475i)5-s − 2.93i·6-s + (−0.992 + 0.125i)7-s + 0.985i·8-s + (−1.15 + 1.99i)9-s + (0.767 + 1.32i)10-s + (0.0805 − 0.0465i)11-s + (1.46 − 2.53i)12-s + (−0.893 + 0.449i)13-s + (−1.48 − 0.626i)14-s − 1.72i·15-s + (0.00874 − 0.0151i)16-s + (−0.215 − 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.991 - 0.131i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.991 - 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49813 + 0.0989566i\)
\(L(\frac12)\) \(\approx\) \(1.49813 + 0.0989566i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 - 0.331i)T \)
13 \( 1 + (3.22 - 1.62i)T \)
good2 \( 1 + (-1.97 - 1.14i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.57 + 2.72i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.84 - 1.06i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.267 + 0.154i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.887 + 1.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.54 - 0.890i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.575 + 0.996i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 + (-3.98 + 2.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.79 - 2.77i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.72iT - 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + (8.24 + 4.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.72 + 6.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.03 - 4.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.72 + 2.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.9 - 6.30i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.96 + 6.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + (-1.43 - 0.829i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.66iT - 97T^{2} \)
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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

−13.74703266819814677226295151271, −13.27703519515757595603882153282, −12.36709260663471887132980006896, −11.62389854300182466241449466340, −9.891440851741504687926580040236, −7.68112892275808858805887736690, −6.52125481117292187174948338630, −6.34559797701970958903569622640, −5.06278720821192302349102056843, −2.64536409751740290627262888545, 3.13343858264985584512889285619, 4.45468948827076162494110134853, 5.37242309085666242706397283004, 6.23805183806654676755336744695, 9.337410209404598618396950349605, 10.05475512804507510055090897969, 10.89497216425471005337525416291, 12.05354758396439577277675144148, 12.83565607729728395323771348172, 13.94354390431974915659755051050

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