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Properties

Label 2-99-33.20-c2-0-4
Degree $2$
Conductor $99$
Sign $0.842 - 0.538i$
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  + (2.84 + 0.925i)2-s + (4.02 + 2.92i)4-s + (1.53 − 0.498i)5-s + (2.69 + 1.95i)7-s + (1.70 + 2.34i)8-s + 4.82·10-s + (−9.45 + 5.61i)11-s + (2.33 − 7.19i)13-s + (5.86 + 8.06i)14-s + (−3.45 − 10.6i)16-s + (−26.9 + 8.76i)17-s + (21.0 − 15.3i)19-s + (7.61 + 2.47i)20-s + (−32.1 + 7.25i)22-s − 19.5i·23-s + ⋯
L(s)  = 1  + (1.42 + 0.462i)2-s + (1.00 + 0.730i)4-s + (0.306 − 0.0996i)5-s + (0.384 + 0.279i)7-s + (0.213 + 0.293i)8-s + 0.482·10-s + (−0.859 + 0.510i)11-s + (0.179 − 0.553i)13-s + (0.418 + 0.576i)14-s + (−0.215 − 0.664i)16-s + (−1.58 + 0.515i)17-s + (1.10 − 0.806i)19-s + (0.380 + 0.123i)20-s + (−1.46 + 0.329i)22-s − 0.850i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.842 - 0.538i$
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: \((2,\ 99,\ (\ :1),\ 0.842 - 0.538i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.54075 + 0.743046i\)
\(L(\frac12)\) \(\approx\) \(2.54075 + 0.743046i\)
\(L(2)\) 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 \( 1 + (9.45 - 5.61i)T \)
good2 \( 1 + (-2.84 - 0.925i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-1.53 + 0.498i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (-2.69 - 1.95i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-2.33 + 7.19i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (26.9 - 8.76i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-21.0 + 15.3i)T + (111. - 343. i)T^{2} \)
23 \( 1 + 19.5iT - 529T^{2} \)
29 \( 1 + (1.26 - 1.74i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (2.85 - 8.77i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-54.6 - 39.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-17.0 - 23.5i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 0.719T + 1.84e3T^{2} \)
47 \( 1 + (-14.4 - 19.8i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (5.50 + 1.78i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (43.4 - 59.8i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-22.3 - 68.6i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 79.8T + 4.48e3T^{2} \)
71 \( 1 + (-102. + 33.3i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-73.1 - 53.1i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-35.4 + 109. i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-119. + 38.8i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 83.3iT - 7.92e3T^{2} \)
97 \( 1 + (24.1 - 74.2i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.47142624099550406007599381184, −13.17250583118167421009190109341, −11.98968396290600810608396534298, −10.85307252267396816230308949987, −9.377840082818503323952278965234, −7.86921756121798056410927899246, −6.57000477887267010368304201345, −5.39631224571052804705334030759, −4.45967410943947026993749420286, −2.67549964708665877447689441284, 2.29877423427251158356596287454, 3.86756239510039142207402562470, 5.12567435108905550924913210505, 6.21371105090830768655942901363, 7.80179274507548214160562410851, 9.391671744014984013374267028633, 10.90344008828359450512325847405, 11.49835953995834082421470153886, 12.71014371674634207140056087910, 13.76113344611608229581237895945

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