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Properties

Label 2-99-9.4-c1-0-9
Degree $2$
Conductor $99$
Sign $-0.766 + 0.642i$
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  + (−0.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 1.73i·6-s + (−2 − 3.46i)7-s − 3·8-s + (1.5 + 2.59i)9-s + 0.999·10-s + (−0.5 − 0.866i)11-s + (−1.5 + 0.866i)12-s + (1 − 1.73i)13-s + (−1.99 + 3.46i)14-s + (1.5 − 0.866i)15-s + (0.500 + 0.866i)16-s + 4·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.707i·6-s + (−0.755 − 1.30i)7-s − 1.06·8-s + (0.5 + 0.866i)9-s + 0.316·10-s + (−0.150 − 0.261i)11-s + (−0.433 + 0.249i)12-s + (0.277 − 0.480i)13-s + (−0.534 + 0.925i)14-s + (0.387 − 0.223i)15-s + (0.125 + 0.216i)16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202460 - 0.556256i\)
\(L(\frac12)\) \(\approx\) \(0.202460 - 0.556256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.27822013136542774411645067193, −12.24770058345093324769664278513, −11.12068507170117195840403650770, −10.54363075271048286899074012374, −9.647883036145881999756952435296, −7.61794058476299345578589135841, −6.69097271344469319803161695803, −5.47409273696594000546430109847, −3.30853676265399904895947095637, −0.904444128019741620890853103834, 3.34313592016290719518536859041, 5.31443056878605563456492793011, 6.25363066566129382179616227129, 7.54137603475492517890220889167, 9.008704682195978642361957307805, 9.664937831690178008006320371290, 11.41821905813335549101636262316, 12.11548685292366160224939635786, 12.85625338382604329221267823853, 14.82018520961101456489570357146

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