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Properties

Label 2-99-9.7-c5-0-17
Degree $2$
Conductor $99$
Sign $0.812 + 0.582i$
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  + (4.21 − 7.30i)2-s + (−13.1 + 8.30i)3-s + (−19.5 − 33.9i)4-s + (4.92 + 8.52i)5-s + (5.02 + 131. i)6-s + (−50.4 + 87.4i)7-s − 60.3·8-s + (105. − 219. i)9-s + 83.0·10-s + (60.5 − 104. i)11-s + (539. + 284. i)12-s + (536. + 929. i)13-s + (425. + 737. i)14-s + (−135. − 71.5i)15-s + (372. − 644. i)16-s + 308.·17-s + ⋯
L(s)  = 1  + (0.745 − 1.29i)2-s + (−0.846 + 0.532i)3-s + (−0.611 − 1.05i)4-s + (0.0880 + 0.152i)5-s + (0.0569 + 1.49i)6-s + (−0.389 + 0.674i)7-s − 0.333·8-s + (0.432 − 0.901i)9-s + 0.262·10-s + (0.150 − 0.261i)11-s + (1.08 + 0.570i)12-s + (0.880 + 1.52i)13-s + (0.580 + 1.00i)14-s + (−0.155 − 0.0821i)15-s + (0.363 − 0.629i)16-s + 0.258·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.812 + 0.582i$
Analytic conductor: \(15.8779\)
Root analytic conductor: \(3.98472\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :5/2),\ 0.812 + 0.582i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.05411 - 0.659898i\)
\(L(\frac12)\) \(\approx\) \(2.05411 - 0.659898i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.1 - 8.30i)T \)
11 \( 1 + (-60.5 + 104. i)T \)
good2 \( 1 + (-4.21 + 7.30i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-4.92 - 8.52i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (50.4 - 87.4i)T + (-8.40e3 - 1.45e4i)T^{2} \)
13 \( 1 + (-536. - 929. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 308.T + 1.41e6T^{2} \)
19 \( 1 - 2.96e3T + 2.47e6T^{2} \)
23 \( 1 + (1.54e3 + 2.67e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (3.12e3 - 5.41e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-3.64e3 - 6.31e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.73e3T + 6.93e7T^{2} \)
41 \( 1 + (-7.58e3 - 1.31e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (3.63e3 - 6.30e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-1.06e4 + 1.85e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 1.58e4T + 4.18e8T^{2} \)
59 \( 1 + (6.90e3 + 1.19e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.52e4 - 2.64e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.37e4 - 2.37e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 4.90e4T + 1.80e9T^{2} \)
73 \( 1 + 4.55e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.54e4 + 4.41e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.56e3 + 2.70e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 9.17e4T + 5.58e9T^{2} \)
97 \( 1 + (4.74e3 - 8.21e3i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.36705997132989336258035881762, −11.84710433095971452516154223002, −10.99726801534768804168859765974, −9.998336980724882451787311000507, −8.992881401374746972070950487390, −6.68890118686826289879884849678, −5.47119115698903966645598352342, −4.25597578827356447900684736888, −3.06108237515141043116766545329, −1.24411852436902580655626459106, 0.941369425436545706219876372343, 3.77473349568502113671509945329, 5.37616125524675279554895247386, 5.93938427881800911227682767361, 7.31147341603754387111927338172, 7.82968423947614975366352128777, 9.853896743110018523967903477643, 11.07277195202441141521709679283, 12.37510260712639517510083254374, 13.46005073261606526407812030861

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