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Properties

Label 2-99-11.6-c2-0-7
Degree $2$
Conductor $99$
Sign $0.669 + 0.742i$
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  + (2.40 − 0.782i)2-s + (1.94 − 1.41i)4-s + (2.61 − 8.03i)5-s + (1.43 + 1.97i)7-s + (−2.36 + 3.25i)8-s − 21.3i·10-s + (−2.51 + 10.7i)11-s + (11.1 − 3.63i)13-s + (4.99 + 3.63i)14-s + (−6.12 + 18.8i)16-s + (1.93 + 0.627i)17-s + (−4.97 + 6.85i)19-s + (−6.29 − 19.3i)20-s + (2.31 + 27.7i)22-s − 41.9·23-s + ⋯
L(s)  = 1  + (1.20 − 0.391i)2-s + (0.487 − 0.354i)4-s + (0.522 − 1.60i)5-s + (0.204 + 0.282i)7-s + (−0.295 + 0.407i)8-s − 2.13i·10-s + (−0.228 + 0.973i)11-s + (0.861 − 0.279i)13-s + (0.357 + 0.259i)14-s + (−0.383 + 1.17i)16-s + (0.113 + 0.0368i)17-s + (−0.262 + 0.360i)19-s + (−0.314 − 0.967i)20-s + (0.105 + 1.26i)22-s − 1.82·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.17666 - 0.968383i\)
\(L(\frac12)\) \(\approx\) \(2.17666 - 0.968383i\)
\(L(2)\) 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 \( 1 + (2.51 - 10.7i)T \)
good2 \( 1 + (-2.40 + 0.782i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (-2.61 + 8.03i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (-1.43 - 1.97i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-11.1 + 3.63i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (-1.93 - 0.627i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (4.97 - 6.85i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 41.9T + 529T^{2} \)
29 \( 1 + (-14.4 - 19.9i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-6.74 - 20.7i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-12.9 + 9.40i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-30.9 + 42.5i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 42.3iT - 1.84e3T^{2} \)
47 \( 1 + (13.0 + 9.51i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-15.3 - 47.1i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-16.1 + 11.7i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (113. + 36.7i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 4.41T + 4.48e3T^{2} \)
71 \( 1 + (1.86 - 5.74i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (3.57 + 4.91i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-98.3 + 31.9i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (28.6 + 9.32i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 60.4T + 7.92e3T^{2} \)
97 \( 1 + (-11.3 - 34.8i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−13.39801142531120797020455266557, −12.39201959170687906514076718909, −12.13667731996879390991490741170, −10.45307631786698004519338572840, −9.079146898576057762166395578922, −8.162450974067974761014514403204, −5.96862912839573288250977797477, −5.07884171309290926363359883636, −4.04225836631480518269243831839, −1.92016025138800898946875211020, 2.85844518675920127057902859034, 4.11396397475665245086650451861, 5.98227335866222557219741240203, 6.39721863935877489828589059965, 7.86921033093696861788227441129, 9.696352664698026010027708639764, 10.80348918777077076198716787576, 11.69988259387855281195292492994, 13.31517861006755444265670543106, 13.87405234136588569898754052788

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