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Properties

Label 2-99-99.32-c1-0-8
Degree $2$
Conductor $99$
Sign $-0.760 + 0.649i$
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  + (−1.10 − 1.91i)2-s + (1.59 − 0.663i)3-s + (−1.44 + 2.50i)4-s + (−2.54 − 1.47i)5-s + (−3.04 − 2.33i)6-s + (1.72 − 0.994i)7-s + 1.97·8-s + (2.11 − 2.12i)9-s + 6.50i·10-s + (0.883 + 3.19i)11-s + (−0.652 + 4.97i)12-s + (−3.09 − 1.78i)13-s + (−3.80 − 2.19i)14-s + (−5.05 − 0.663i)15-s + (0.705 + 1.22i)16-s + 6.08·17-s + ⋯
L(s)  = 1  + (−0.782 − 1.35i)2-s + (0.923 − 0.382i)3-s + (−0.723 + 1.25i)4-s + (−1.13 − 0.657i)5-s + (−1.24 − 0.951i)6-s + (0.650 − 0.375i)7-s + 0.699·8-s + (0.706 − 0.707i)9-s + 2.05i·10-s + (0.266 + 0.963i)11-s + (−0.188 + 1.43i)12-s + (−0.857 − 0.495i)13-s + (−1.01 − 0.587i)14-s + (−1.30 − 0.171i)15-s + (0.176 + 0.305i)16-s + 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.760 + 0.649i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.760 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274842 - 0.745301i\)
\(L(\frac12)\) \(\approx\) \(0.274842 - 0.745301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.59 + 0.663i)T \)
11 \( 1 + (-0.883 - 3.19i)T \)
good2 \( 1 + (1.10 + 1.91i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.54 + 1.47i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.72 + 0.994i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (3.09 + 1.78i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.08T + 17T^{2} \)
19 \( 1 - 0.896iT - 19T^{2} \)
23 \( 1 + (-4.90 - 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.60 + 2.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.278 - 0.482i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 + (4.52 - 7.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.14 + 1.24i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.77 - 5.06i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.18iT - 53T^{2} \)
59 \( 1 + (1.19 + 0.689i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.99 - 5.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.471 - 0.815i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.55iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (-10.4 + 6.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.60 + 4.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (1.79 + 3.11i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.95778581726129911119497867208, −12.27288582114348672229116182821, −11.55364399286147206000506694510, −10.11412929716810977310477537714, −9.280833988723052961201390879373, −8.018042551392198047565069188691, −7.57410594778792299728504446933, −4.50121380116984045861615953095, −3.18553724706327534802810977232, −1.34372905792014729267093573512, 3.32077659299677244833915917325, 5.11105177319963837940084317540, 6.91979668108890690323729906926, 7.79064775759433597560744423082, 8.494543303500989045368764821511, 9.514242735009545533739141691199, 10.86742264969333367391862462099, 12.09876066017138890628123344529, 14.01137053957140585681777981431, 14.81901877341470751713045121986

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