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Properties

Label 2-99-9.4-c5-0-30
Degree $2$
Conductor $99$
Sign $-0.669 - 0.742i$
Analytic cond. $15.8779$
Root an. cond. $3.98472$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.11 + 8.86i)2-s + (14.9 − 4.30i)3-s + (−36.3 + 62.9i)4-s + (20.1 − 34.9i)5-s + (114. + 110. i)6-s + (111. + 193. i)7-s − 416.·8-s + (206. − 128. i)9-s + 412.·10-s + (60.5 + 104. i)11-s + (−273. + 1.09e3i)12-s + (231. − 400. i)13-s + (−1.14e3 + 1.97e3i)14-s + (152. − 610. i)15-s + (−968. − 1.67e3i)16-s − 734.·17-s + ⋯
L(s)  = 1  + (0.904 + 1.56i)2-s + (0.961 − 0.275i)3-s + (−1.13 + 1.96i)4-s + (0.360 − 0.625i)5-s + (1.30 + 1.25i)6-s + (0.860 + 1.48i)7-s − 2.30·8-s + (0.847 − 0.530i)9-s + 1.30·10-s + (0.150 + 0.261i)11-s + (−0.549 + 2.20i)12-s + (0.379 − 0.656i)13-s + (−1.55 + 2.69i)14-s + (0.174 − 0.700i)15-s + (−0.945 − 1.63i)16-s − 0.616·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(15.8779\)
Root analytic conductor: \(3.98472\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :5/2),\ -0.669 - 0.742i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.72329 + 3.87355i\)
\(L(\frac12)\) \(\approx\) \(1.72329 + 3.87355i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-14.9 + 4.30i)T \)
11 \( 1 + (-60.5 - 104. i)T \)
good2 \( 1 + (-5.11 - 8.86i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-20.1 + 34.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-111. - 193. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
13 \( 1 + (-231. + 400. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 734.T + 1.41e6T^{2} \)
19 \( 1 + 1.93e3T + 2.47e6T^{2} \)
23 \( 1 + (1.01e3 - 1.75e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (2.88e3 + 4.98e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-2.63e3 + 4.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.50e4T + 6.93e7T^{2} \)
41 \( 1 + (-639. + 1.10e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (3.68e3 + 6.38e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (8.36e3 + 1.44e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 1.00e4T + 4.18e8T^{2} \)
59 \( 1 + (-1.94e4 + 3.36e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.88e4 - 3.26e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.87e3 + 1.53e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 5.76e4T + 1.80e9T^{2} \)
73 \( 1 + 1.97e4T + 2.07e9T^{2} \)
79 \( 1 + (3.91e4 + 6.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (1.45e4 + 2.52e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 1.80e4T + 5.58e9T^{2} \)
97 \( 1 + (-5.96e4 - 1.03e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.33060775789802788904506266737, −12.98262734269866488569243428542, −11.79425785697479461400167871463, −9.338167188683117975352916187672, −8.478164815364341792683597167285, −7.87396352401628509382948751775, −6.33100578636317022186627792007, −5.33664851712264930510065245148, −4.12095820440196724235835538479, −2.23299630539858506166972135170, 1.33425773143700716721668720719, 2.52891589566566201832306075548, 3.94037815040201794966536481769, 4.56962925955502848094510957926, 6.70917010618608447324866983713, 8.428440243190802647466250875756, 9.805786532166085852546120416292, 10.72882927464001621492254211433, 11.13141356606915526810001366173, 12.81505312516490489559818520373

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