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Properties

Label 2-99-11.5-c1-0-1
Degree $2$
Conductor $99$
Sign $0.213 - 0.976i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
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  + (0.476 + 1.46i)2-s + (−0.309 + 0.224i)4-s + (−0.476 + 1.46i)5-s + (−0.190 + 0.138i)7-s + (2.02 + 1.46i)8-s − 2.38·10-s + (−2.31 − 2.37i)11-s + (−1.30 − 4.02i)13-s + (−0.294 − 0.214i)14-s + (−1.42 + 4.39i)16-s + (1.83 − 5.65i)17-s + (3.73 + 2.71i)19-s + (−0.182 − 0.560i)20-s + (2.38 − 4.53i)22-s − 7.49·23-s + ⋯
L(s)  = 1  + (0.337 + 1.03i)2-s + (−0.154 + 0.112i)4-s + (−0.213 + 0.656i)5-s + (−0.0721 + 0.0524i)7-s + (0.714 + 0.518i)8-s − 0.753·10-s + (−0.698 − 0.716i)11-s + (−0.363 − 1.11i)13-s + (−0.0787 − 0.0572i)14-s + (−0.356 + 1.09i)16-s + (0.445 − 1.37i)17-s + (0.857 + 0.622i)19-s + (−0.0407 − 0.125i)20-s + (0.507 − 0.965i)22-s − 1.56·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.213 - 0.976i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.213 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936216 + 0.753354i\)
\(L(\frac12)\) \(\approx\) \(0.936216 + 0.753354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (2.31 + 2.37i)T \)
good2 \( 1 + (-0.476 - 1.46i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.476 - 1.46i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.190 - 0.138i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.30 + 4.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.83 + 5.65i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.73 - 2.71i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.49T + 23T^{2} \)
29 \( 1 + (-1.54 + 1.12i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.736 + 2.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.04 - 3.66i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.51 - 3.28i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (-0.771 - 0.560i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.79 - 8.59i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.83 - 4.96i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.33 + 4.11i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.85T + 67T^{2} \)
71 \( 1 + (-2.38 + 7.33i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.23 + 3.80i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.163 + 0.502i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.36 - 4.18i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (3.95 + 12.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.20755569879558816348627991673, −13.56506545277694676155186049083, −12.06478239331470348338738834125, −10.89979490009467039655098838499, −9.918579959745643594456935926133, −8.054603710489082542636485817051, −7.42889820063496557347167234716, −6.07301908876095571532798415771, −5.11402252571010367400531264764, −3.05229173116034381947425397531, 1.97544366425961921616310001188, 3.80777498854078035110093893622, 4.99066353992131693364620669113, 6.93270139749146723038610449064, 8.216857159549490818395632936840, 9.691902637042252784717179567929, 10.56724531234688140896988090091, 11.86458414854459115168721301095, 12.39706289365213656627170801740, 13.32809481313283364871461346843

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