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Properties

Label 2-99-99.65-c1-0-5
Degree $2$
Conductor $99$
Sign $0.993 + 0.112i$
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  + (1.18 − 1.26i)3-s + (1 + 1.73i)4-s + (−0.813 + 0.469i)5-s + (−0.186 − 2.99i)9-s + (−2.87 − 1.65i)11-s + (3.37 + 0.792i)12-s + (−0.372 + 1.58i)15-s + (−1.99 + 3.46i)16-s + (−1.62 − 0.939i)20-s + (−2.87 + 1.65i)23-s + (−2.05 + 3.56i)25-s + (−4.00 − 3.31i)27-s + (−3.05 − 5.29i)31-s + (−5.5 + 1.65i)33-s + (5 − 3.31i)36-s + 12.1·37-s + ⋯
L(s)  = 1  + (0.684 − 0.728i)3-s + (0.5 + 0.866i)4-s + (−0.363 + 0.210i)5-s + (−0.0620 − 0.998i)9-s + (−0.866 − 0.500i)11-s + (0.973 + 0.228i)12-s + (−0.0961 + 0.409i)15-s + (−0.499 + 0.866i)16-s + (−0.363 − 0.210i)20-s + (−0.598 + 0.345i)23-s + (−0.411 + 0.713i)25-s + (−0.769 − 0.638i)27-s + (−0.549 − 0.951i)31-s + (−0.957 + 0.288i)33-s + (0.833 − 0.552i)36-s + 1.99·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.993 + 0.112i$
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: \((2,\ 99,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20739 - 0.0679551i\)
\(L(\frac12)\) \(\approx\) \(1.20739 - 0.0679551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 + 1.26i)T \)
11 \( 1 + (2.87 + 1.65i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.813 - 0.469i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (2.87 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.05 + 5.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.8 - 6.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 + (-12.6 + 7.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.55 + 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (0.0584 - 0.101i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.63046764088606868027637931815, −12.92268541914789442581456543554, −11.90584862539892331619490590227, −10.97256984907390893482249688000, −9.280744764461936783193494757493, −7.959651358483919217713316062134, −7.53587474141776881299877334320, −6.11549310294296475393631446842, −3.76428668241376040415798472380, −2.50649730523536048982707790685, 2.45479558764738213060773335021, 4.35755257875907660958260563927, 5.60928290974810850798941750951, 7.31394114624828432805182236509, 8.497247340343113472245905424425, 9.820656808171214247429748722585, 10.47341347487820387673598975424, 11.60675812704474031125500806808, 13.01767498681225041307164392941, 14.23096617009642410908557449294

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