Properties

Label 2-99-33.5-c2-0-5
Degree $2$
Conductor $99$
Sign $-0.355 + 0.934i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.355 + 0.934i$
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: \((2,\ 99,\ (\ :1),\ -0.355 + 0.934i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.103478 - 0.150024i\)
\(L(\frac12)\) \(\approx\) \(0.103478 - 0.150024i\)
\(L(2)\) 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

−13.29484253464962946152628584971, −11.98590671646268549765760032432, −10.76062147791561339595030318092, −9.782739401118262329203669803094, −8.928565729522436432005813266066, −8.022942639220572315714425321415, −6.64038614453960653858347612209, −5.78635443118238907231713567115, −2.74130430247590810781141276365, −0.22379913558127552558320330181, 1.97220992083253777981309265106, 3.89206378231432860312893279631, 6.80598980789728808956977814510, 7.24201186665146499825303346946, 8.956866534107135870483078148782, 9.585210000119431104177977704032, 10.47053760639144974479682546911, 11.55673331708431930492804332931, 12.53732728843712239256144839384, 13.79750799757518500622468073878

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