Properties

Label 2-983-983.2-c1-0-34
Degree $2$
Conductor $983$
Sign $-0.954 - 0.298i$
Analytic cond. $7.84929$
Root an. cond. $2.80165$
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  + (−0.815 + 2.43i)2-s + (1.99 + 1.11i)3-s + (−3.66 − 2.76i)4-s + (0.269 − 0.407i)5-s + (−4.34 + 3.93i)6-s + (0.302 − 0.667i)7-s + (5.48 − 3.76i)8-s + (1.15 + 1.89i)9-s + (0.771 + 0.988i)10-s + (4.53 + 0.290i)11-s + (−4.20 − 9.61i)12-s + (0.900 + 1.04i)13-s + (1.37 + 1.27i)14-s + (0.993 − 0.509i)15-s + (2.16 + 7.58i)16-s + (−3.64 + 1.60i)17-s + ⋯
L(s)  = 1  + (−0.576 + 1.72i)2-s + (1.15 + 0.646i)3-s + (−1.83 − 1.38i)4-s + (0.120 − 0.182i)5-s + (−1.77 + 1.60i)6-s + (0.114 − 0.252i)7-s + (1.93 − 1.32i)8-s + (0.385 + 0.633i)9-s + (0.243 + 0.312i)10-s + (1.36 + 0.0876i)11-s + (−1.21 − 2.77i)12-s + (0.249 + 0.288i)13-s + (0.368 + 0.341i)14-s + (0.256 − 0.131i)15-s + (0.541 + 1.89i)16-s + (−0.883 + 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(983\)
Sign: $-0.954 - 0.298i$
Analytic conductor: \(7.84929\)
Root analytic conductor: \(2.80165\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{983} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 983,\ (\ :1/2),\ -0.954 - 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256711 + 1.68236i\)
\(L(\frac12)\) \(\approx\) \(0.256711 + 1.68236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad983 \( 1 + (-18.9 - 24.9i)T \)
good2 \( 1 + (0.815 - 2.43i)T + (-1.59 - 1.20i)T^{2} \)
3 \( 1 + (-1.99 - 1.11i)T + (1.56 + 2.56i)T^{2} \)
5 \( 1 + (-0.269 + 0.407i)T + (-1.94 - 4.60i)T^{2} \)
7 \( 1 + (-0.302 + 0.667i)T + (-4.61 - 5.26i)T^{2} \)
11 \( 1 + (-4.53 - 0.290i)T + (10.9 + 1.40i)T^{2} \)
13 \( 1 + (-0.900 - 1.04i)T + (-1.86 + 12.8i)T^{2} \)
17 \( 1 + (3.64 - 1.60i)T + (11.4 - 12.5i)T^{2} \)
19 \( 1 + (-3.65 - 2.79i)T + (4.98 + 18.3i)T^{2} \)
23 \( 1 + (-0.465 - 4.39i)T + (-22.4 + 4.82i)T^{2} \)
29 \( 1 + (0.311 - 0.450i)T + (-10.2 - 27.1i)T^{2} \)
31 \( 1 + (-0.0749 - 0.111i)T + (-11.7 + 28.7i)T^{2} \)
37 \( 1 + (4.98 - 1.23i)T + (32.7 - 17.2i)T^{2} \)
41 \( 1 + (-1.33 - 0.120i)T + (40.3 + 7.30i)T^{2} \)
43 \( 1 + (-2.35 - 3.23i)T + (-13.1 + 40.9i)T^{2} \)
47 \( 1 + (2.50 + 8.37i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (-2.40 + 2.20i)T + (4.57 - 52.8i)T^{2} \)
59 \( 1 + (13.7 + 1.05i)T + (58.3 + 9.02i)T^{2} \)
61 \( 1 + (0.904 + 0.767i)T + (9.90 + 60.1i)T^{2} \)
67 \( 1 + (-7.74 + 6.66i)T + (10.0 - 66.2i)T^{2} \)
71 \( 1 + (5.15 - 4.20i)T + (14.2 - 69.5i)T^{2} \)
73 \( 1 + (-9.34 - 6.68i)T + (23.6 + 69.0i)T^{2} \)
79 \( 1 + (-3.78 + 0.560i)T + (75.6 - 22.9i)T^{2} \)
83 \( 1 + (-0.781 + 0.242i)T + (68.4 - 46.9i)T^{2} \)
89 \( 1 + (-5.71 - 3.76i)T + (35.1 + 81.7i)T^{2} \)
97 \( 1 + (5.66 - 0.290i)T + (96.4 - 9.91i)T^{2} \)
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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

−9.600939562879895889247747361555, −9.345326266991517893969676377323, −8.703074478547614136529344054807, −7.979195265587244494065745855829, −7.11304382961621220910783121408, −6.36227583604904710580858988973, −5.32916088253938846132424306320, −4.28602553769022889788128343727, −3.52705417764941343620100714647, −1.47565834719340534481561285813, 0.968297217043251410731599743750, 2.09797016874898160003554125611, 2.80389524768495728970678328318, 3.67878330256501913372941715660, 4.71571641423138959071113455020, 6.45880055053483172653873158454, 7.45968002014112104544855511771, 8.455383155220300371927617128684, 8.977384145852221444776199347295, 9.409795062426625454262309445924

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