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Properties

Label 2-99-11.7-c2-0-4
Degree $2$
Conductor $99$
Sign $0.636 - 0.771i$
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  + (−1.30 + 1.79i)2-s + (−0.280 − 0.862i)4-s + (7.03 − 5.11i)5-s + (6.34 − 2.06i)7-s + (−6.51 − 2.11i)8-s + 19.2i·10-s + (8.95 + 6.38i)11-s + (−6.55 + 9.02i)13-s + (−4.56 + 14.0i)14-s + (15.2 − 11.0i)16-s + (4.60 + 6.33i)17-s + (−8.02 − 2.60i)19-s + (−6.37 − 4.63i)20-s + (−23.1 + 7.72i)22-s − 9.30·23-s + ⋯
L(s)  = 1  + (−0.651 + 0.896i)2-s + (−0.0700 − 0.215i)4-s + (1.40 − 1.02i)5-s + (0.906 − 0.294i)7-s + (−0.814 − 0.264i)8-s + 1.92i·10-s + (0.813 + 0.580i)11-s + (−0.504 + 0.694i)13-s + (−0.326 + 1.00i)14-s + (0.950 − 0.690i)16-s + (0.270 + 0.372i)17-s + (−0.422 − 0.137i)19-s + (−0.318 − 0.231i)20-s + (−1.05 + 0.351i)22-s − 0.404·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.636 - 0.771i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.636 - 0.771i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13472 + 0.535189i\)
\(L(\frac12)\) \(\approx\) \(1.13472 + 0.535189i\)
\(L(2)\) 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 \( 1 + (-8.95 - 6.38i)T \)
good2 \( 1 + (1.30 - 1.79i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-7.03 + 5.11i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (-6.34 + 2.06i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (6.55 - 9.02i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (-4.60 - 6.33i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (8.02 + 2.60i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 9.30T + 529T^{2} \)
29 \( 1 + (6.82 - 2.21i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-22.1 - 16.1i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (16.3 + 50.3i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (45.5 + 14.7i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 45.8iT - 1.84e3T^{2} \)
47 \( 1 + (-4.96 + 15.2i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (44.4 + 32.2i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-22.0 - 67.7i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (0.764 + 1.05i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 28.9T + 4.48e3T^{2} \)
71 \( 1 + (-17.8 + 12.9i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (119. - 38.6i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-25.2 + 34.6i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (78.3 + 107. i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 9.48T + 7.92e3T^{2} \)
97 \( 1 + (126. + 91.7i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−14.07834389947698418353657170980, −12.78438680954878553636974899473, −11.85412382588500935261205534097, −10.09842294994873652054120994335, −9.200206538709575937985941251377, −8.418269827504502056666170527395, −7.07043571533182924178428094902, −5.91338598917504360769176364097, −4.59788945321436904959010724844, −1.69570336579125726284097529453, 1.69465603591320646236938764469, 2.93058498765515956254175562131, 5.47392175604739647046790005172, 6.51717413240842480632766134694, 8.316036469502196635315882292472, 9.548875456499836525742164121718, 10.26397044376316643877816649230, 11.15623142266230906665470845851, 12.05163499973468405965211553861, 13.62160808752666128346562357710

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