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Properties

Label 2-99-9.4-c1-0-1
Degree $2$
Conductor $99$
Sign $-0.500 - 0.866i$
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  + (0.939 + 1.62i)2-s + (−1.70 + 0.300i)3-s + (−0.766 + 1.32i)4-s + (−1.43 + 2.49i)5-s + (−2.09 − 2.49i)6-s + (0.326 + 0.565i)7-s + 0.879·8-s + (2.81 − 1.02i)9-s − 5.41·10-s + (0.5 + 0.866i)11-s + (0.907 − 2.49i)12-s + (3.37 − 5.85i)13-s + (−0.613 + 1.06i)14-s + (1.70 − 4.68i)15-s + (2.35 + 4.08i)16-s + 0.184·17-s + ⋯
L(s)  = 1  + (0.664 + 1.15i)2-s + (−0.984 + 0.173i)3-s + (−0.383 + 0.663i)4-s + (−0.643 + 1.11i)5-s + (−0.854 − 1.01i)6-s + (0.123 + 0.213i)7-s + 0.310·8-s + (0.939 − 0.342i)9-s − 1.71·10-s + (0.150 + 0.261i)11-s + (0.262 − 0.719i)12-s + (0.937 − 1.62i)13-s + (−0.163 + 0.283i)14-s + (0.440 − 1.21i)15-s + (0.589 + 1.02i)16-s + 0.0448·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518567 + 0.898185i\)
\(L(\frac12)\) \(\approx\) \(0.518567 + 0.898185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.300i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.939 - 1.62i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.43 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.326 - 0.565i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-3.37 + 5.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.184T + 17T^{2} \)
19 \( 1 + 5.22T + 19T^{2} \)
23 \( 1 + (-1.59 + 2.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 - 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.553 - 0.957i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.106T + 37T^{2} \)
41 \( 1 + (-2.80 + 4.86i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.92 + 3.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.00 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + (5.27 - 9.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.67 - 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.90 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (-0.733 - 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.520 - 0.902i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.01T + 89T^{2} \)
97 \( 1 + (-2.86 - 4.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.75648955819797762141413907904, −13.29674243050505664512120908319, −12.31907917792042465022701315280, −10.86059941714504891459072775709, −10.51179848334022285439567020510, −8.266052948927218701638265756799, −7.07013772828168894407102789621, −6.28886996829218224912112058240, −5.17843851421376892230602417199, −3.73666614660862836087454104850, 1.43255679275398639033860616590, 4.07484953111814333481348863115, 4.72213850669722601074317444081, 6.39750311496769766351948640263, 8.033488627298117695249270513392, 9.499955869669467671586087336823, 11.08948993537939147894747004982, 11.40381364776942494347732484418, 12.43760275348914562073493384079, 13.06057713531468769412059313647

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