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Properties

Label 2-99-11.5-c3-0-13
Degree $2$
Conductor $99$
Sign $-0.962 - 0.271i$
Analytic cond. $5.84118$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.38 − 4.25i)2-s + (−9.70 + 7.05i)4-s + (5.80 − 17.8i)5-s + (23.2 − 16.8i)7-s + (14.4 + 10.5i)8-s − 84.0·10-s + (−27.5 + 23.8i)11-s + (−0.802 − 2.46i)13-s + (−103. − 75.4i)14-s + (−4.94 + 15.2i)16-s + (−4.39 + 13.5i)17-s + (8.24 + 5.98i)19-s + (69.7 + 214. i)20-s + (139. + 84.2i)22-s + 86.1·23-s + ⋯
L(s)  = 1  + (−0.488 − 1.50i)2-s + (−1.21 + 0.881i)4-s + (0.519 − 1.59i)5-s + (1.25 − 0.911i)7-s + (0.639 + 0.464i)8-s − 2.65·10-s + (−0.755 + 0.654i)11-s + (−0.0171 − 0.0526i)13-s + (−1.98 − 1.44i)14-s + (−0.0772 + 0.237i)16-s + (−0.0627 + 0.193i)17-s + (0.0995 + 0.0723i)19-s + (0.779 + 2.39i)20-s + (1.35 + 0.816i)22-s + 0.781·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.962 - 0.271i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ -0.962 - 0.271i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.168857 + 1.22229i\)
\(L(\frac12)\) \(\approx\) \(0.168857 + 1.22229i\)
\(L(\frac{5}{2})\) 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 \( 1 + (27.5 - 23.8i)T \)
good2 \( 1 + (1.38 + 4.25i)T + (-6.47 + 4.70i)T^{2} \)
5 \( 1 + (-5.80 + 17.8i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (-23.2 + 16.8i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (0.802 + 2.46i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (4.39 - 13.5i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-8.24 - 5.98i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 - 86.1T + 1.21e4T^{2} \)
29 \( 1 + (120. - 87.3i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (24.9 + 76.6i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-187. + 136. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (-66.3 - 48.2i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 60.9T + 7.95e4T^{2} \)
47 \( 1 + (-89.8 - 65.2i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (101. + 313. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-563. + 409. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (82.0 - 252. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 - 664.T + 3.00e5T^{2} \)
71 \( 1 + (271. - 834. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (438. - 318. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (330. + 1.01e3i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-222. + 683. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 1.52e3T + 7.04e5T^{2} \)
97 \( 1 + (-257. - 791. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−12.86147890053853729456208305989, −11.64797632192695335855391969878, −10.71875489541378027501182347652, −9.726106163588416536694856776699, −8.756755256884316659183447612388, −7.73088833540136019954945570395, −5.19023628552979712634494809798, −4.21477629729229186898182796066, −1.96699085742203328292713961345, −0.878251673783393472631306028583, 2.60629485880533139144703870636, 5.24310719833056289278475775248, 6.11267755392868009882691547632, 7.28299144094181210141194223420, 8.178209822589938414777963507457, 9.320292323870440352889368431953, 10.68729331530551198890237145850, 11.56005845302572760408848264673, 13.54150901587204769238992856788, 14.45940754140499035481451667545

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