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Properties

Label 2-99-9.7-c1-0-1
Degree $2$
Conductor $99$
Sign $-0.500 - 0.866i$
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  + (−0.766 + 1.32i)2-s + (0.592 + 1.62i)3-s + (−0.173 − 0.300i)4-s + (0.266 + 0.460i)5-s + (−2.61 − 0.460i)6-s + (1.43 − 2.49i)7-s − 2.53·8-s + (−2.29 + 1.92i)9-s − 0.815·10-s + (0.5 − 0.866i)11-s + (0.386 − 0.460i)12-s + (−0.0320 − 0.0555i)13-s + (2.20 + 3.82i)14-s + (−0.592 + 0.705i)15-s + (2.28 − 3.96i)16-s + 1.22·17-s + ⋯
L(s)  = 1  + (−0.541 + 0.938i)2-s + (0.342 + 0.939i)3-s + (−0.0868 − 0.150i)4-s + (0.118 + 0.206i)5-s + (−1.06 − 0.188i)6-s + (0.544 − 0.942i)7-s − 0.895·8-s + (−0.766 + 0.642i)9-s − 0.257·10-s + (0.150 − 0.261i)11-s + (0.111 − 0.133i)12-s + (−0.00889 − 0.0154i)13-s + (0.589 + 1.02i)14-s + (−0.152 + 0.182i)15-s + (0.571 − 0.990i)16-s + 0.297·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447513 + 0.775116i\)
\(L(\frac12)\) \(\approx\) \(0.447513 + 0.775116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.592 - 1.62i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.766 - 1.32i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.266 - 0.460i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.43 + 2.49i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.0320 + 0.0555i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.22T + 17T^{2} \)
19 \( 1 - 0.411T + 19T^{2} \)
23 \( 1 + (-2.11 - 3.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.16 + 7.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.97 - 6.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.94T + 37T^{2} \)
41 \( 1 + (4.46 + 7.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.71 - 9.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.92 - 3.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.448T + 53T^{2} \)
59 \( 1 + (6.84 + 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.56 + 4.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.92 + 8.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + 8.96T + 73T^{2} \)
79 \( 1 + (-1.32 + 2.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.81 - 4.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.97T + 89T^{2} \)
97 \( 1 + (-4.95 + 8.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.41124815076799714032386837984, −13.77629534406165829688857543566, −11.95214984641441554716136391913, −10.75838368415143326318162928417, −9.797300586650818190305322940457, −8.594192364028186558476982941750, −7.76827769790549747607730812702, −6.48801023790524987408416705268, −4.91702843983405519247896985251, −3.31033569234948605735922775028, 1.57470559883870459294167480629, 2.90466115875686840687765127827, 5.44036792417210398304367993872, 6.82326055833356684491144028129, 8.446514738846544850645845869214, 9.041960345084908878729025972509, 10.38883095267806002082387049519, 11.71593511000459598578647675288, 12.15745897031685128017206330216, 13.25195723496314944452163680451

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