Properties

Label 2-99-99.58-c1-0-4
Degree $2$
Conductor $99$
Sign $-0.329 + 0.944i$
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  + (−2.39 − 1.06i)2-s + (0.360 − 1.69i)3-s + (3.24 + 3.60i)4-s + (2.95 − 1.31i)5-s + (−2.66 + 3.66i)6-s + (0.978 − 0.207i)7-s + (−2.30 − 7.10i)8-s + (−2.74 − 1.22i)9-s − 8.47·10-s + (−2.38 + 2.30i)11-s + (7.28 − 4.20i)12-s + (−0.258 − 2.45i)13-s + (−2.56 − 0.544i)14-s + (−1.16 − 5.48i)15-s + (−1.03 + 9.80i)16-s + (1.5 + 1.08i)17-s + ⋯
L(s)  = 1  + (−1.69 − 0.752i)2-s + (0.207 − 0.978i)3-s + (1.62 + 1.80i)4-s + (1.32 − 0.588i)5-s + (−1.08 + 1.49i)6-s + (0.369 − 0.0785i)7-s + (−0.816 − 2.51i)8-s + (−0.913 − 0.406i)9-s − 2.67·10-s + (−0.717 + 0.696i)11-s + (2.10 − 1.21i)12-s + (−0.0716 − 0.681i)13-s + (−0.684 − 0.145i)14-s + (−0.300 − 1.41i)15-s + (−0.257 + 2.45i)16-s + (0.363 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351709 - 0.495302i\)
\(L(\frac12)\) \(\approx\) \(0.351709 - 0.495302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.360 + 1.69i)T \)
11 \( 1 + (2.38 - 2.30i)T \)
good2 \( 1 + (2.39 + 1.06i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (-2.95 + 1.31i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (-0.978 + 0.207i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (0.258 + 2.45i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-1.5 - 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.19 - 2.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.81 - 0.385i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.169 - 1.60i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-1.88 + 5.79i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-11.2 - 2.38i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.42 - 4.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.97 - 4.41i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-5.73 + 4.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.413 + 0.459i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.433 - 4.12i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.3 - 8.28i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.11 - 6.51i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.73 + 3.88i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-1.48 + 14.1i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-5.48 - 2.44i)T + (64.9 + 72.0i)T^{2} \)
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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

−13.00580953597126243180880204158, −12.57996847248017842786876847558, −11.24819015708460243145797846702, −10.09655023060878248103820480215, −9.307893521162440938250664734089, −8.194441375470245257483212747697, −7.38836171951280720471647718363, −5.78262692902104318883977465513, −2.55966398499463621512819833532, −1.40426069247133108431275174212, 2.39534913888368332658669380092, 5.39467360116086427635480141781, 6.36981426161494575469330794501, 7.86266675230813577644449648855, 8.976034656473504248295718813515, 9.743674920337018349641530924504, 10.52060011433752043937799261572, 11.29668953224283084280513483841, 13.78962407554264520287355011027, 14.52792480560273714358527364016

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