Properties

Label 2-99-33.20-c2-0-5
Degree $2$
Conductor $99$
Sign $0.0217 + 0.999i$
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  + (−0.296 − 0.0962i)2-s + (−3.15 − 2.29i)4-s + (5.65 − 1.83i)5-s + (−7.01 − 5.09i)7-s + (1.44 + 1.99i)8-s − 1.85·10-s + (0.122 − 10.9i)11-s + (5.77 − 17.7i)13-s + (1.58 + 2.18i)14-s + (4.58 + 14.1i)16-s + (9.12 − 2.96i)17-s + (−4.66 + 3.38i)19-s + (−22.0 − 7.17i)20-s + (−1.09 + 3.24i)22-s + 41.5i·23-s + ⋯
L(s)  = 1  + (−0.148 − 0.0481i)2-s + (−0.789 − 0.573i)4-s + (1.13 − 0.367i)5-s + (−1.00 − 0.728i)7-s + (0.180 + 0.248i)8-s − 0.185·10-s + (0.0111 − 0.999i)11-s + (0.444 − 1.36i)13-s + (0.113 + 0.156i)14-s + (0.286 + 0.882i)16-s + (0.537 − 0.174i)17-s + (−0.245 + 0.178i)19-s + (−1.10 − 0.358i)20-s + (−0.0497 + 0.147i)22-s + 1.80i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.0217 + 0.999i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.0217 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.763473 - 0.747010i\)
\(L(\frac12)\) \(\approx\) \(0.763473 - 0.747010i\)
\(L(2)\) 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 \( 1 + (-0.122 + 10.9i)T \)
good2 \( 1 + (0.296 + 0.0962i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-5.65 + 1.83i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (7.01 + 5.09i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-5.77 + 17.7i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-9.12 + 2.96i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (4.66 - 3.38i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 41.5iT - 529T^{2} \)
29 \( 1 + (-10.2 + 14.0i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (4.22 - 13.0i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-5.83 - 4.24i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-31.2 - 42.9i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 43.3T + 1.84e3T^{2} \)
47 \( 1 + (11.0 + 15.2i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-51.8 - 16.8i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-20.7 + 28.5i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (36.4 + 112. i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 91.5T + 4.48e3T^{2} \)
71 \( 1 + (-110. + 35.7i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-42.8 - 31.1i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-0.633 + 1.94i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (27.4 - 8.90i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 134. iT - 7.92e3T^{2} \)
97 \( 1 + (3.08 - 9.48i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.48925133420175105712412505508, −12.81258894211082900818878317598, −10.89209997221895752775338228797, −9.957654815042901352386830332175, −9.348348557800190609435618095198, −8.015194944138607528384135832163, −6.12180903510129025287882601554, −5.41151737024088111197429518695, −3.48965580825623057709370547072, −0.935423601826079165951985261084, 2.48087177054465348230640656599, 4.29425942243133982943928621089, 5.95862361243567408146981499040, 7.01851685580166934003543206121, 8.793540409798303466810270317432, 9.448775781961744259567753269426, 10.35636380331815322568814283008, 12.17418665699626595955666905689, 12.83601747926115431492214521689, 13.86466029209222689648894770727

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