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Properties

Label 2-99-11.6-c2-0-2
Degree $2$
Conductor $99$
Sign $0.263 - 0.964i$
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  + (0.690 − 0.224i)2-s + (−2.80 + 2.04i)4-s + (−1.23 + 3.80i)5-s + (5.85 + 8.05i)7-s + (−3.19 + 4.39i)8-s + 2.90i·10-s + (10.3 − 3.66i)11-s + (−5 + 1.62i)13-s + (5.85 + 4.25i)14-s + (3.07 − 9.45i)16-s + (−14.5 − 4.72i)17-s + (1.21 − 1.67i)19-s + (−4.29 − 13.2i)20-s + (6.34 − 4.86i)22-s + 2.76·23-s + ⋯
L(s)  = 1  + (0.345 − 0.112i)2-s + (−0.702 + 0.510i)4-s + (−0.247 + 0.760i)5-s + (0.836 + 1.15i)7-s + (−0.398 + 0.549i)8-s + 0.290i·10-s + (0.942 − 0.333i)11-s + (−0.384 + 0.124i)13-s + (0.418 + 0.303i)14-s + (0.192 − 0.591i)16-s + (−0.854 − 0.277i)17-s + (0.0641 − 0.0882i)19-s + (−0.214 − 0.660i)20-s + (0.288 − 0.220i)22-s + 0.120·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.263 - 0.964i$
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: \((2,\ 99,\ (\ :1),\ 0.263 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05395 + 0.804703i\)
\(L(\frac12)\) \(\approx\) \(1.05395 + 0.804703i\)
\(L(2)\) 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 \( 1 + (-10.3 + 3.66i)T \)
good2 \( 1 + (-0.690 + 0.224i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (1.23 - 3.80i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (-5.85 - 8.05i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (5 - 1.62i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (14.5 + 4.72i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-1.21 + 1.67i)T + (-111. - 343. i)T^{2} \)
23 \( 1 - 2.76T + 529T^{2} \)
29 \( 1 + (-16.7 - 22.9i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (2.20 + 6.77i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-32.5 + 23.6i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-41.2 + 56.7i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 23.0iT - 1.84e3T^{2} \)
47 \( 1 + (-22.0 - 16.0i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-3.54 - 10.8i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (1.83 - 1.33i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-21.5 - 6.98i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 38.4T + 4.48e3T^{2} \)
71 \( 1 + (23.5 - 72.5i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-60.4 - 83.2i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (3.74 - 1.21i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (79.1 + 25.7i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 123.T + 7.92e3T^{2} \)
97 \( 1 + (23.9 + 73.6i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−14.11289524739449259959010642921, −12.70864027413919258129973955123, −11.79712490879193417055109961991, −11.05104982021057452557668042877, −9.239901454824583546600538650882, −8.543594126062576285124486479048, −7.14146280274915968879203629287, −5.55740318747259502300861822026, −4.22561177673324350550488588067, −2.66584417720853587645212967447, 1.05348301032202285128618022536, 4.21517623471232129988026357820, 4.74858927655265091212621922335, 6.44898807642749248458862382829, 7.927680020653157223492857228386, 9.062151626062364994708372187058, 10.14680935593644218263423261116, 11.36748206523416126393598191810, 12.59823272129965386418011373808, 13.54087297433234645965617517195

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