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Properties

Label 2-99-99.32-c1-0-5
Degree $2$
Conductor $99$
Sign $0.348 - 0.937i$
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  + (1.10 + 1.91i)2-s + (1.59 − 0.663i)3-s + (−1.44 + 2.50i)4-s + (−2.54 − 1.47i)5-s + (3.04 + 2.33i)6-s + (−1.72 + 0.994i)7-s − 1.97·8-s + (2.11 − 2.12i)9-s − 6.50i·10-s + (−2.32 − 2.36i)11-s + (−0.652 + 4.97i)12-s + (3.09 + 1.78i)13-s + (−3.80 − 2.19i)14-s + (−5.05 − 0.663i)15-s + (0.705 + 1.22i)16-s − 6.08·17-s + ⋯
L(s)  = 1  + (0.782 + 1.35i)2-s + (0.923 − 0.382i)3-s + (−0.723 + 1.25i)4-s + (−1.13 − 0.657i)5-s + (1.24 + 0.951i)6-s + (−0.650 + 0.375i)7-s − 0.699·8-s + (0.706 − 0.707i)9-s − 2.05i·10-s + (−0.701 − 0.712i)11-s + (−0.188 + 1.43i)12-s + (0.857 + 0.495i)13-s + (−1.01 − 0.587i)14-s + (−1.30 − 0.171i)15-s + (0.176 + 0.305i)16-s − 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.348 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24428 + 0.864616i\)
\(L(\frac12)\) \(\approx\) \(1.24428 + 0.864616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.59 + 0.663i)T \)
11 \( 1 + (2.32 + 2.36i)T \)
good2 \( 1 + (-1.10 - 1.91i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.54 + 1.47i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.72 - 0.994i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-3.09 - 1.78i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.08T + 17T^{2} \)
19 \( 1 + 0.896iT - 19T^{2} \)
23 \( 1 + (-4.90 - 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.60 - 2.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.278 - 0.482i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 + (-4.52 + 7.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.14 - 1.24i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.77 - 5.06i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.18iT - 53T^{2} \)
59 \( 1 + (1.19 + 0.689i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.99 + 5.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.471 - 0.815i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.55iT - 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + (10.4 - 6.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.60 - 4.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (1.79 + 3.11i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.12773047546168165118085467460, −13.18659089073359331429502276193, −12.72777417033238579309369409221, −11.20803341865488198541699541818, −8.988524393084725224685273609184, −8.394693336771034110922780611126, −7.32664300207950566784059465334, −6.27537115472086887741443817207, −4.63421164814362728315293929104, −3.42416178019835276501783845108, 2.65029653164336333691362939876, 3.65153467300332753276550732098, 4.59933612211086575259469826242, 7.02187986502418798851538512081, 8.295261314034900427385291868434, 9.842548800198200789632296732510, 10.69698332066614296463295923006, 11.44306891930330991240801865589, 13.01490055553511698250759849429, 13.21048654391714217247694072359

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