Properties

Label 2-983-983.2-c1-0-22
Degree $2$
Conductor $983$
Sign $0.784 - 0.619i$
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  + (−0.691 + 2.06i)2-s + (−1.31 − 0.740i)3-s + (−2.18 − 1.65i)4-s + (−2.30 + 3.47i)5-s + (2.43 − 2.20i)6-s + (0.363 − 0.802i)7-s + (1.33 − 0.915i)8-s + (−0.373 − 0.612i)9-s + (−5.57 − 7.15i)10-s + (−1.49 − 0.0955i)11-s + (1.66 + 3.79i)12-s + (0.264 + 0.306i)13-s + (1.40 + 1.30i)14-s + (5.60 − 2.87i)15-s + (−0.538 − 1.88i)16-s + (−0.645 + 0.285i)17-s + ⋯
L(s)  = 1  + (−0.488 + 1.46i)2-s + (−0.760 − 0.427i)3-s + (−1.09 − 0.825i)4-s + (−1.02 + 1.55i)5-s + (0.995 − 0.901i)6-s + (0.137 − 0.303i)7-s + (0.471 − 0.323i)8-s + (−0.124 − 0.204i)9-s + (−1.76 − 2.26i)10-s + (−0.449 − 0.0288i)11-s + (0.479 + 1.09i)12-s + (0.0734 + 0.0848i)13-s + (0.375 + 0.349i)14-s + (1.44 − 0.741i)15-s + (−0.134 − 0.471i)16-s + (−0.156 + 0.0691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(983\)
Sign: $0.784 - 0.619i$
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: \((2,\ 983,\ (\ :1/2),\ 0.784 - 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309345 + 0.107433i\)
\(L(\frac12)\) \(\approx\) \(0.309345 + 0.107433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad983 \( 1 + (31.3 - 1.54i)T \)
good2 \( 1 + (0.691 - 2.06i)T + (-1.59 - 1.20i)T^{2} \)
3 \( 1 + (1.31 + 0.740i)T + (1.56 + 2.56i)T^{2} \)
5 \( 1 + (2.30 - 3.47i)T + (-1.94 - 4.60i)T^{2} \)
7 \( 1 + (-0.363 + 0.802i)T + (-4.61 - 5.26i)T^{2} \)
11 \( 1 + (1.49 + 0.0955i)T + (10.9 + 1.40i)T^{2} \)
13 \( 1 + (-0.264 - 0.306i)T + (-1.86 + 12.8i)T^{2} \)
17 \( 1 + (0.645 - 0.285i)T + (11.4 - 12.5i)T^{2} \)
19 \( 1 + (4.63 + 3.54i)T + (4.98 + 18.3i)T^{2} \)
23 \( 1 + (-0.765 - 7.22i)T + (-22.4 + 4.82i)T^{2} \)
29 \( 1 + (-5.18 + 7.49i)T + (-10.2 - 27.1i)T^{2} \)
31 \( 1 + (-0.728 - 1.08i)T + (-11.7 + 28.7i)T^{2} \)
37 \( 1 + (3.37 - 0.838i)T + (32.7 - 17.2i)T^{2} \)
41 \( 1 + (-3.64 - 0.327i)T + (40.3 + 7.30i)T^{2} \)
43 \( 1 + (1.36 + 1.87i)T + (-13.1 + 40.9i)T^{2} \)
47 \( 1 + (1.87 + 6.26i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (6.30 - 5.78i)T + (4.57 - 52.8i)T^{2} \)
59 \( 1 + (4.44 + 0.341i)T + (58.3 + 9.02i)T^{2} \)
61 \( 1 + (-5.53 - 4.69i)T + (9.90 + 60.1i)T^{2} \)
67 \( 1 + (-6.71 + 5.77i)T + (10.0 - 66.2i)T^{2} \)
71 \( 1 + (2.96 - 2.42i)T + (14.2 - 69.5i)T^{2} \)
73 \( 1 + (-11.9 - 8.57i)T + (23.6 + 69.0i)T^{2} \)
79 \( 1 + (-1.80 + 0.267i)T + (75.6 - 22.9i)T^{2} \)
83 \( 1 + (-7.05 + 2.18i)T + (68.4 - 46.9i)T^{2} \)
89 \( 1 + (-5.07 - 3.33i)T + (35.1 + 81.7i)T^{2} \)
97 \( 1 + (-10.7 + 0.549i)T + (96.4 - 9.91i)T^{2} \)
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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

−10.09738163616239197594864765005, −8.950243830475223812995666097117, −7.949830046040144890896267772060, −7.47380123732908890413759642490, −6.60746124857414749496124874762, −6.39316455728316930290834761493, −5.25833213631130041257551478330, −4.01130057424110293484044110861, −2.75968719114508741545075882472, −0.29857205933894798669662779313, 0.78281440496656498650795152698, 2.17828800367097125162399963971, 3.59930554210483276465016927618, 4.59260134910802528144326673001, 5.03143604598121129286798653134, 6.31698594016167386509842707542, 8.017556802896626080106496395695, 8.470419688011341552784498077830, 9.117993724205675387320028564254, 10.21243439394444103516314287340

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