Properties

Label 2-91-91.31-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.0968 + 0.995i$
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  + (−0.746 − 0.200i)2-s + (−0.421 − 0.243i)3-s + (−1.21 − 0.701i)4-s + (0.472 − 1.76i)5-s + (0.266 + 0.266i)6-s + (−0.210 − 2.63i)7-s + (1.85 + 1.85i)8-s + (−1.38 − 2.39i)9-s + (−0.705 + 1.22i)10-s + (0.990 − 0.265i)11-s + (0.341 + 0.591i)12-s + (−0.266 + 3.59i)13-s + (−0.370 + 2.01i)14-s + (−0.628 + 0.628i)15-s + (0.386 + 0.669i)16-s + (2.60 − 4.50i)17-s + ⋯
L(s)  = 1  + (−0.527 − 0.141i)2-s + (−0.243 − 0.140i)3-s + (−0.607 − 0.350i)4-s + (0.211 − 0.788i)5-s + (0.108 + 0.108i)6-s + (−0.0796 − 0.996i)7-s + (0.657 + 0.657i)8-s + (−0.460 − 0.797i)9-s + (−0.222 + 0.386i)10-s + (0.298 − 0.0800i)11-s + (0.0986 + 0.170i)12-s + (−0.0738 + 0.997i)13-s + (−0.0989 + 0.537i)14-s + (−0.162 + 0.162i)15-s + (0.0966 + 0.167i)16-s + (0.630 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0968 + 0.995i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0968 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409766 - 0.451577i\)
\(L(\frac12)\) \(\approx\) \(0.409766 - 0.451577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.210 + 2.63i)T \)
13 \( 1 + (0.266 - 3.59i)T \)
good2 \( 1 + (0.746 + 0.200i)T + (1.73 + i)T^{2} \)
3 \( 1 + (0.421 + 0.243i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.472 + 1.76i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.990 + 0.265i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.60 + 4.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.36 - 5.07i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.730 + 0.421i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + (5.69 - 1.52i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.61 + 6.03i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.0927 + 0.0927i)T + 41iT^{2} \)
43 \( 1 + 7.36iT - 43T^{2} \)
47 \( 1 + (-2.17 - 0.583i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.38 + 5.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.60 - 9.73i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.13 + 0.653i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.11 - 4.16i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.02 - 6.02i)T - 71iT^{2} \)
73 \( 1 + (-2.93 - 10.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.16 - 8.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.16 + 4.16i)T + 83iT^{2} \)
89 \( 1 + (7.49 + 2.00i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.49 - 2.49i)T + 97iT^{2} \)
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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

−14.00998077558890050122206823884, −12.71191820220181739684305870597, −11.65133394677447277056633416735, −10.34342127924307845413148614012, −9.378600360207768182485916068368, −8.548656571483139079232375125808, −6.99649301417255582390982921848, −5.45403370585516561370357061800, −4.10062245946867137866254355658, −1.02509958508267018611401192096, 2.93215372098286006262286611485, 4.93673584719689706704388867656, 6.32135201731032512167399312230, 7.88641280324703509084661225926, 8.763156701757725204306879671257, 10.04093586633991189954592313937, 10.90386495849704973061700958326, 12.29915657195273366294204215095, 13.28448265147358339694040831825, 14.45611605116729390029734075522

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