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Properties

Label 2-99-99.65-c1-0-1
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: $\chi_{99} (65, \cdot )$
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

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

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