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Properties

Label 2-99-11.3-c1-0-3
Degree $2$
Conductor $99$
Sign $0.503 + 0.863i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 1.26i)2-s + (0.809 − 2.48i)4-s + (−1.73 − 1.26i)5-s + (−1.30 + 4.02i)7-s + (−0.410 − 1.26i)8-s − 4.61·10-s + (3.22 − 0.780i)11-s + (−0.190 + 0.138i)13-s + (2.81 + 8.65i)14-s + (1.92 + 1.40i)16-s + (−4.96 − 3.60i)17-s + (−0.736 − 2.26i)19-s + (−4.55 + 3.30i)20-s + (4.61 − 5.42i)22-s − 3.98·23-s + ⋯
L(s)  = 1  + (1.22 − 0.893i)2-s + (0.404 − 1.24i)4-s + (−0.777 − 0.564i)5-s + (−0.494 + 1.52i)7-s + (−0.145 − 0.446i)8-s − 1.46·10-s + (0.971 − 0.235i)11-s + (−0.0529 + 0.0384i)13-s + (0.751 + 2.31i)14-s + (0.481 + 0.350i)16-s + (−1.20 − 0.874i)17-s + (−0.168 − 0.519i)19-s + (−1.01 + 0.739i)20-s + (0.984 − 1.15i)22-s − 0.830·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.503 + 0.863i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36828 - 0.785978i\)
\(L(\frac12)\) \(\approx\) \(1.36828 - 0.785978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.22 + 0.780i)T \)
good2 \( 1 + (-1.73 + 1.26i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.73 + 1.26i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.30 - 4.02i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.190 - 0.138i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.96 + 3.60i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.736 + 2.26i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.98T + 23T^{2} \)
29 \( 1 + (2.14 - 6.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.73 + 2.71i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.545 + 1.67i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.917 - 2.82i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.70T + 43T^{2} \)
47 \( 1 + (1.07 + 3.30i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.48 - 1.07i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.30 + 7.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-9.16 - 6.65i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.85T + 67T^{2} \)
71 \( 1 + (-8.69 - 6.31i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.763 + 2.35i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.66 + 5.56i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.70 - 4.86i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.90T + 89T^{2} \)
97 \( 1 + (9.54 - 6.93i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.45297624432555476747512971621, −12.47437564043447823023468068140, −11.88250238556484323842356550803, −11.19383106626660836763305764001, −9.404966983106112268637873990902, −8.463924397990291598615711481885, −6.43910264623084145944378702522, −5.14152842929907872875689142425, −3.96728289859875483370294704413, −2.48707916072600681071858994349, 3.76683826341343913067536598885, 4.23769931370944255059205384103, 6.27730222672653421828775026783, 6.97487737602903230110780174224, 7.962801359668442935634200495726, 9.921948183733247603026942734782, 11.09771681756174527903447661819, 12.30637447213800432926540179140, 13.37405598131043456316398258738, 14.10201071241340869490623790259

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