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Properties

Label 2-99-11.9-c3-0-12
Degree $2$
Conductor $99$
Sign $-0.938 - 0.345i$
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  + (1.45 − 4.49i)2-s + (−11.5 − 8.41i)4-s + (−1.86 − 5.72i)5-s + (−8.05 − 5.85i)7-s + (−24.1 + 17.5i)8-s − 28.4·10-s + (−8.30 − 35.5i)11-s + (−27.8 + 85.8i)13-s + (−38.0 + 27.6i)14-s + (8.12 + 24.9i)16-s + (−13.8 − 42.6i)17-s + (110. − 80.5i)19-s + (−26.6 + 81.9i)20-s + (−171. − 14.5i)22-s + 71.4·23-s + ⋯
L(s)  = 1  + (0.516 − 1.58i)2-s + (−1.44 − 1.05i)4-s + (−0.166 − 0.512i)5-s + (−0.435 − 0.316i)7-s + (−1.06 + 0.774i)8-s − 0.899·10-s + (−0.227 − 0.973i)11-s + (−0.594 + 1.83i)13-s + (−0.726 + 0.527i)14-s + (0.126 + 0.390i)16-s + (−0.197 − 0.608i)17-s + (1.33 − 0.972i)19-s + (−0.297 + 0.915i)20-s + (−1.66 − 0.140i)22-s + 0.647·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ -0.938 - 0.345i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.260273 + 1.46013i\)
\(L(\frac12)\) \(\approx\) \(0.260273 + 1.46013i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.30 + 35.5i)T \)
good2 \( 1 + (-1.45 + 4.49i)T + (-6.47 - 4.70i)T^{2} \)
5 \( 1 + (1.86 + 5.72i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (8.05 + 5.85i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (27.8 - 85.8i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (13.8 + 42.6i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-110. + 80.5i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 71.4T + 1.21e4T^{2} \)
29 \( 1 + (119. + 86.8i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-2.38 + 7.34i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (8.18 + 5.94i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-305. + 222. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 276.T + 7.95e4T^{2} \)
47 \( 1 + (191. - 139. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (-68.6 + 211. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (-136. - 99.4i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-70.1 - 215. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 + 362.T + 3.00e5T^{2} \)
71 \( 1 + (-219. - 676. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-845. - 614. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-15.6 + 48.3i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-378. - 1.16e3i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + 964.T + 7.04e5T^{2} \)
97 \( 1 + (-78.8 + 242. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−12.71452171089888093413450567482, −11.63305654673600183355195159717, −11.12170359247418035451034101086, −9.659469442578136556134416836778, −9.021464637953347797837761034306, −7.06816044626571483923261861095, −5.15684958019998542526963556810, −4.03493265994379916747170754346, −2.61875115088741162010280590723, −0.74378002485790855893083522004, 3.26520788980237775332053493239, 5.00966396561350161213845980224, 5.96510484010246028989639855754, 7.29512129126418028656815688599, 7.88035620145052150939422459865, 9.423567662662898274473844047248, 10.66115255466831835830628946537, 12.48832494315645348657590122704, 13.03671081555851196923460813161, 14.44173708229508693419765642129

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