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Properties

Label 2-99-99.65-c1-0-4
Degree $2$
Conductor $99$
Sign $0.985 + 0.172i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.679 − 1.17i)2-s + (−0.323 + 1.70i)3-s + (0.0778 + 0.134i)4-s + (0.901 − 0.520i)5-s + (1.78 + 1.53i)6-s + (−0.600 − 0.346i)7-s + 2.92·8-s + (−2.79 − 1.10i)9-s − 1.41i·10-s + (0.484 − 3.28i)11-s + (−0.254 + 0.0888i)12-s + (−5.22 + 3.01i)13-s + (−0.816 + 0.471i)14-s + (0.593 + 1.70i)15-s + (1.83 − 3.17i)16-s − 3.56·17-s + ⋯
L(s)  = 1  + (0.480 − 0.831i)2-s + (−0.186 + 0.982i)3-s + (0.0389 + 0.0673i)4-s + (0.403 − 0.232i)5-s + (0.727 + 0.626i)6-s + (−0.227 − 0.131i)7-s + 1.03·8-s + (−0.930 − 0.366i)9-s − 0.446i·10-s + (0.146 − 0.989i)11-s + (−0.0734 + 0.0256i)12-s + (−1.44 + 0.836i)13-s + (−0.218 + 0.125i)14-s + (0.153 + 0.439i)15-s + (0.458 − 0.793i)16-s − 0.864·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.985 + 0.172i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25957 - 0.109255i\)
\(L(\frac12)\) \(\approx\) \(1.25957 - 0.109255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.323 - 1.70i)T \)
11 \( 1 + (-0.484 + 3.28i)T \)
good2 \( 1 + (-0.679 + 1.17i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.901 + 0.520i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.600 + 0.346i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (5.22 - 3.01i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
23 \( 1 + (-4.37 + 2.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.54 - 2.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0574 - 0.0995i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.416T + 37T^{2} \)
41 \( 1 + (1.90 + 3.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.79 - 4.49i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.70 - 2.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.94iT - 53T^{2} \)
59 \( 1 + (8.91 - 5.14i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.23 - 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.48iT - 71T^{2} \)
73 \( 1 - 9.05iT - 73T^{2} \)
79 \( 1 + (-0.482 - 0.278i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.98 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.48iT - 89T^{2} \)
97 \( 1 + (5.39 - 9.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.74795718568249133755668293602, −12.73794038884474849473103784034, −11.56627646520048683407018341819, −10.93290619612618391797134522395, −9.761918167363311161363179803674, −8.795475626643610095658044149325, −6.95583233471710039304659453722, −5.23189910988341356828596976430, −4.12444749338138119808692412994, −2.68745849625470951858547991505, 2.23664832891507374795330959984, 4.90746889805627704243256606525, 6.02922399224402823641149688255, 7.03052615560339775148035069240, 7.79985350208736333892216734909, 9.619776920669324987416522491584, 10.78196215538478288794521910339, 12.21030868026546972567168671571, 13.02113893910598636391789535459, 14.02633793107485097303118525952

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