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Properties

Label 2-99-11.8-c2-0-7
Degree $2$
Conductor $99$
Sign $-0.292 + 0.956i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.184 + 0.253i)2-s + (1.20 − 3.71i)4-s + (−5.99 − 4.35i)5-s + (−9.53 − 3.09i)7-s + (2.35 − 0.764i)8-s − 2.32i·10-s + (10.2 + 3.93i)11-s + (2.00 + 2.75i)13-s + (−0.970 − 2.98i)14-s + (−12.0 − 8.71i)16-s + (9.14 − 12.5i)17-s + (29.1 − 9.46i)19-s + (−23.4 + 17.0i)20-s + (0.892 + 3.32i)22-s − 7.67·23-s + ⋯
L(s)  = 1  + (0.0920 + 0.126i)2-s + (0.301 − 0.927i)4-s + (−1.19 − 0.871i)5-s + (−1.36 − 0.442i)7-s + (0.294 − 0.0955i)8-s − 0.232i·10-s + (0.933 + 0.358i)11-s + (0.153 + 0.211i)13-s + (−0.0692 − 0.213i)14-s + (−0.750 − 0.544i)16-s + (0.537 − 0.740i)17-s + (1.53 − 0.498i)19-s + (−1.17 + 0.850i)20-s + (0.0405 + 0.151i)22-s − 0.333·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.292 + 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.591459 - 0.799352i\)
\(L(\frac12)\) \(\approx\) \(0.591459 - 0.799352i\)
\(L(2)\) 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 \( 1 + (-10.2 - 3.93i)T \)
good2 \( 1 + (-0.184 - 0.253i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (5.99 + 4.35i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (9.53 + 3.09i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-2.00 - 2.75i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-9.14 + 12.5i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-29.1 + 9.46i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 7.67T + 529T^{2} \)
29 \( 1 + (3.21 + 1.04i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-3.27 + 2.37i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (0.734 - 2.25i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (6.69 - 2.17i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 3.99iT - 1.84e3T^{2} \)
47 \( 1 + (15.2 + 47.0i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (48.3 - 35.1i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-3.39 + 10.4i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-43.6 + 60.1i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 3.22T + 4.48e3T^{2} \)
71 \( 1 + (-94.5 - 68.7i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-17.8 - 5.78i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (2.06 + 2.84i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-86.4 + 119. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 65.8T + 7.92e3T^{2} \)
97 \( 1 + (52.4 - 38.1i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−13.38814044853123369795309748695, −12.15335924569693313499900560078, −11.43802156853490151145405127026, −9.895157002550571972076352140944, −9.209555463560971190045675080823, −7.51000356417973213988827887224, −6.54282015485629150350174166905, −4.99603077826731721775667161302, −3.61263745200706383897746362953, −0.76838909336429150128664928597, 3.18868001478285922683736664916, 3.70484588017860947664009859519, 6.21828553886804076117593164849, 7.24828600478985377792127189667, 8.261673202516210725635976624855, 9.670236885739123874382742987378, 11.07652546547650421370854925783, 11.95709388763401841562564497526, 12.58199208824053681597698285782, 13.91970120928359891215327212033

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