Properties

Label 2-983-983.492-c1-0-35
Degree $2$
Conductor $983$
Sign $-0.911 + 0.411i$
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  + (−0.805 − 2.40i)2-s + (−0.153 + 0.0861i)3-s + (−3.54 + 2.67i)4-s + (−0.680 − 1.02i)5-s + (0.330 + 0.299i)6-s + (1.03 + 2.28i)7-s + (5.09 + 3.49i)8-s + (−1.54 + 2.53i)9-s + (−1.92 + 2.46i)10-s + (−2.29 + 0.147i)11-s + (0.313 − 0.715i)12-s + (3.73 − 4.31i)13-s + (4.65 − 4.32i)14-s + (0.192 + 0.0987i)15-s + (1.87 − 6.54i)16-s + (0.975 + 0.430i)17-s + ⋯
L(s)  = 1  + (−0.569 − 1.70i)2-s + (−0.0885 + 0.0497i)3-s + (−1.77 + 1.33i)4-s + (−0.304 − 0.458i)5-s + (0.135 + 0.122i)6-s + (0.391 + 0.863i)7-s + (1.80 + 1.23i)8-s + (−0.514 + 0.845i)9-s + (−0.607 + 0.779i)10-s + (−0.693 + 0.0444i)11-s + (0.0903 − 0.206i)12-s + (1.03 − 1.19i)13-s + (1.24 − 1.15i)14-s + (0.0497 + 0.0255i)15-s + (0.467 − 1.63i)16-s + (0.236 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(983\)
Sign: $-0.911 + 0.411i$
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: \((2,\ 983,\ (\ :1/2),\ -0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170566 - 0.791849i\)
\(L(\frac12)\) \(\approx\) \(0.170566 - 0.791849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad983 \( 1 + (-15.7 + 27.0i)T \)
good2 \( 1 + (0.805 + 2.40i)T + (-1.59 + 1.20i)T^{2} \)
3 \( 1 + (0.153 - 0.0861i)T + (1.56 - 2.56i)T^{2} \)
5 \( 1 + (0.680 + 1.02i)T + (-1.94 + 4.60i)T^{2} \)
7 \( 1 + (-1.03 - 2.28i)T + (-4.61 + 5.26i)T^{2} \)
11 \( 1 + (2.29 - 0.147i)T + (10.9 - 1.40i)T^{2} \)
13 \( 1 + (-3.73 + 4.31i)T + (-1.86 - 12.8i)T^{2} \)
17 \( 1 + (-0.975 - 0.430i)T + (11.4 + 12.5i)T^{2} \)
19 \( 1 + (-0.853 + 0.652i)T + (4.98 - 18.3i)T^{2} \)
23 \( 1 + (0.310 - 2.92i)T + (-22.4 - 4.82i)T^{2} \)
29 \( 1 + (2.97 + 4.30i)T + (-10.2 + 27.1i)T^{2} \)
31 \( 1 + (-4.90 + 7.30i)T + (-11.7 - 28.7i)T^{2} \)
37 \( 1 + (-1.22 - 0.303i)T + (32.7 + 17.2i)T^{2} \)
41 \( 1 + (8.48 - 0.762i)T + (40.3 - 7.30i)T^{2} \)
43 \( 1 + (-3.51 + 4.82i)T + (-13.1 - 40.9i)T^{2} \)
47 \( 1 + (-1.40 + 4.68i)T + (-39.2 - 25.8i)T^{2} \)
53 \( 1 + (-1.18 - 1.08i)T + (4.57 + 52.8i)T^{2} \)
59 \( 1 + (-8.33 + 0.641i)T + (58.3 - 9.02i)T^{2} \)
61 \( 1 + (-1.43 + 1.22i)T + (9.90 - 60.1i)T^{2} \)
67 \( 1 + (-3.86 - 3.32i)T + (10.0 + 66.2i)T^{2} \)
71 \( 1 + (3.48 + 2.84i)T + (14.2 + 69.5i)T^{2} \)
73 \( 1 + (3.87 - 2.76i)T + (23.6 - 69.0i)T^{2} \)
79 \( 1 + (-9.82 - 1.45i)T + (75.6 + 22.9i)T^{2} \)
83 \( 1 + (2.89 + 0.898i)T + (68.4 + 46.9i)T^{2} \)
89 \( 1 + (2.23 - 1.46i)T + (35.1 - 81.7i)T^{2} \)
97 \( 1 + (-2.47 - 0.126i)T + (96.4 + 9.91i)T^{2} \)
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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

−9.936961008520632119493773250120, −8.796141082595147915595232979301, −8.286544702965635346293106302695, −7.81662387619998332471329736018, −5.78041231832392999846653390490, −5.09786476315039813944996638777, −3.95720602964027211036859360934, −2.86314779000101078458027223390, −2.06496987271433280160770617829, −0.58051289949415841007762679631, 1.07805204899359639181296886453, 3.39407473027739473009204536156, 4.47870626924605759569453517857, 5.46575203723212707818724853321, 6.46281297064172369942425721068, 6.92613331026135332422894836650, 7.71714034074771098854512785862, 8.567387225408221882524206994429, 9.116353407926264915270726703087, 10.16541340910437683774813942536

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