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Properties

Label 2-99-11.4-c1-0-2
Degree $2$
Conductor $99$
Sign $-0.503 + 0.863i$
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  + (−1.73 − 1.26i)2-s + (0.809 + 2.48i)4-s + (1.73 − 1.26i)5-s + (−1.30 − 4.02i)7-s + (0.410 − 1.26i)8-s − 4.61·10-s + (−3.22 − 0.780i)11-s + (−0.190 − 0.138i)13-s + (−2.81 + 8.65i)14-s + (1.92 − 1.40i)16-s + (4.96 − 3.60i)17-s + (−0.736 + 2.26i)19-s + (4.55 + 3.30i)20-s + (4.61 + 5.42i)22-s + 3.98·23-s + ⋯
L(s)  = 1  + (−1.22 − 0.893i)2-s + (0.404 + 1.24i)4-s + (0.777 − 0.564i)5-s + (−0.494 − 1.52i)7-s + (0.145 − 0.446i)8-s − 1.46·10-s + (−0.971 − 0.235i)11-s + (−0.0529 − 0.0384i)13-s + (−0.751 + 2.31i)14-s + (0.481 − 0.350i)16-s + (1.20 − 0.874i)17-s + (−0.168 + 0.519i)19-s + (1.01 + 0.739i)20-s + (0.984 + 1.15i)22-s + 0.830·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.503 + 0.863i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283149 - 0.492927i\)
\(L(\frac12)\) \(\approx\) \(0.283149 - 0.492927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (3.22 + 0.780i)T \)
good2 \( 1 + (1.73 + 1.26i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-1.73 + 1.26i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.30 + 4.02i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.190 + 0.138i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.96 + 3.60i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.736 - 2.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.98T + 23T^{2} \)
29 \( 1 + (-2.14 - 6.61i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.73 - 2.71i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.545 - 1.67i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.917 - 2.82i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.70T + 43T^{2} \)
47 \( 1 + (-1.07 + 3.30i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.48 - 1.07i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.30 + 7.09i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.16 + 6.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.85T + 67T^{2} \)
71 \( 1 + (8.69 - 6.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.763 - 2.35i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.66 - 5.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.70 - 4.86i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.90T + 89T^{2} \)
97 \( 1 + (9.54 + 6.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.33874906993058910753262045499, −12.40330400800730717211146210368, −10.97689273109095656707271204215, −10.17249319789931601031644645537, −9.582289154190474838956331460133, −8.277959486294670864035204387520, −7.16745378410721064430473741566, −5.24932177805359400080755508986, −3.13160208856049619505524521450, −1.07313258794831466183241338671, 2.57871003985398342127592857640, 5.62712327188243204286702406973, 6.33610167744056927485100486823, 7.72223354493215271637297475809, 8.788966779431701074935743152579, 9.748337247681020656371448084062, 10.47922753757290378885639170099, 12.16049313117143654646547169054, 13.25113797350179124783858718583, 14.79955660120720480422558176444

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