Properties

Label 2-983-983.492-c1-0-54
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: $\chi_{983} (492, \cdot )$
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

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

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