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Properties

Label 2-99-33.5-c2-0-7
Degree $2$
Conductor $99$
Sign $0.762 + 0.646i$
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  + (3.44 − 1.11i)2-s + (7.36 − 5.35i)4-s + (0.157 + 0.0510i)5-s + (−8.33 + 6.05i)7-s + (10.8 − 14.9i)8-s + 0.598·10-s + (−3.56 + 10.4i)11-s + (−6.29 − 19.3i)13-s + (−21.9 + 30.1i)14-s + (9.42 − 29.0i)16-s + (21.8 + 7.11i)17-s + (3.34 + 2.43i)19-s + (1.43 − 0.464i)20-s + (−0.641 + 39.8i)22-s + 13.5i·23-s + ⋯
L(s)  = 1  + (1.72 − 0.559i)2-s + (1.84 − 1.33i)4-s + (0.0314 + 0.0102i)5-s + (−1.19 + 0.864i)7-s + (1.35 − 1.86i)8-s + 0.0598·10-s + (−0.324 + 0.945i)11-s + (−0.483 − 1.48i)13-s + (−1.56 + 2.15i)14-s + (0.589 − 1.81i)16-s + (1.28 + 0.418i)17-s + (0.176 + 0.128i)19-s + (0.0715 − 0.0232i)20-s + (−0.0291 + 1.80i)22-s + 0.587i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.762 + 0.646i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.762 + 0.646i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.76852 - 1.01531i\)
\(L(\frac12)\) \(\approx\) \(2.76852 - 1.01531i\)
\(L(2)\) 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 \( 1 + (3.56 - 10.4i)T \)
good2 \( 1 + (-3.44 + 1.11i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (-0.157 - 0.0510i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (8.33 - 6.05i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (6.29 + 19.3i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (-21.8 - 7.11i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-3.34 - 2.43i)T + (111. + 343. i)T^{2} \)
23 \( 1 - 13.5iT - 529T^{2} \)
29 \( 1 + (2.36 + 3.24i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (6.33 + 19.4i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (35.2 - 25.6i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-30.6 + 42.1i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 62.5T + 1.84e3T^{2} \)
47 \( 1 + (-22.7 + 31.3i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (33.6 - 10.9i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (11.5 + 15.9i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (1.98 - 6.10i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 3.98T + 4.48e3T^{2} \)
71 \( 1 + (-52.0 - 16.9i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-18.4 + 13.4i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (17.6 + 54.3i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (54.0 + 17.5i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 28.1iT - 7.92e3T^{2} \)
97 \( 1 + (-25.6 - 78.8i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−13.26235483110896554549508352868, −12.44994129660021866293895200261, −12.12005446420747676593527195105, −10.47983714799034480921136070573, −9.715818240129685552694995186765, −7.56179126712968571044459505325, −6.01436038422584607034678428052, −5.32100159270818420963691870998, −3.61458931854475048862784356259, −2.49406139900327818541848393053, 3.07256298853268789463071666745, 4.14583772590717311509462145666, 5.60756471849628428801797713906, 6.68418938790816700002057069443, 7.55766088381461919689368367662, 9.494236453095228696533429152988, 10.96997224978683104292996183713, 12.11340527208789971639763510778, 12.95848620768528818830899200401, 13.98893241477669036927543291263

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