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Properties

Label 2-99-33.32-c1-0-3
Degree $2$
Conductor $99$
Sign $0.997 + 0.0659i$
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  + 1.73·2-s + 0.999·4-s − 1.41i·5-s + 2.44i·7-s − 1.73·8-s − 2.44i·10-s + (−1.73 − 2.82i)11-s + 4.89i·13-s + 4.24i·14-s − 5·16-s − 7.34i·19-s − 1.41i·20-s + (−2.99 − 4.89i)22-s + 2.82i·23-s + 2.99·25-s + 8.48i·26-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s − 0.632i·5-s + 0.925i·7-s − 0.612·8-s − 0.774i·10-s + (−0.522 − 0.852i)11-s + 1.35i·13-s + 1.13i·14-s − 1.25·16-s − 1.68i·19-s − 0.316i·20-s + (−0.639 − 1.04i)22-s + 0.589i·23-s + 0.599·25-s + 1.66i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.997 + 0.0659i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.997 + 0.0659i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58916 - 0.0524713i\)
\(L(\frac12)\) \(\approx\) \(1.58916 - 0.0524713i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2.82i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 2.44iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 9.89iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 4.89iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 12.2iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−13.68642115610286660432311105395, −13.04359172855784564928119747738, −11.98080252994570488079927553944, −11.25033661634587718559180118348, −9.282488771616602212298812887487, −8.610143931275478110278774821258, −6.65327855109073829931799890249, −5.44408447847065810026241746411, −4.51514897553382018020429493092, −2.80058170977831227301855613318, 2.99297515248484655175350708632, 4.28026523619816053340617922409, 5.59184807281069130468428611784, 6.88648207920864990292020820073, 8.098278463691958770896283990155, 10.01958857385252337075061154060, 10.72374507669906852458487536860, 12.24895702682139049913130344600, 12.91221870578719934605318588220, 13.95844459681045818018906594389

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