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Properties

Label 2-783-9.4-c1-0-17
Degree $2$
Conductor $783$
Sign $0.806 + 0.591i$
Analytic cond. $6.25228$
Root an. cond. $2.50045$
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  + (−0.100 − 0.173i)2-s + (0.979 − 1.69i)4-s + (−0.556 + 0.964i)5-s + (0.359 + 0.622i)7-s − 0.793·8-s + 0.223·10-s + (0.874 + 1.51i)11-s + (0.587 − 1.01i)13-s + (0.0719 − 0.124i)14-s + (−1.88 − 3.25i)16-s + 4.38·17-s + 4.97·19-s + (1.09 + 1.88i)20-s + (0.175 − 0.303i)22-s + (3.61 − 6.26i)23-s + ⋯
L(s)  = 1  + (−0.0708 − 0.122i)2-s + (0.489 − 0.848i)4-s + (−0.248 + 0.431i)5-s + (0.135 + 0.235i)7-s − 0.280·8-s + 0.0705·10-s + (0.263 + 0.456i)11-s + (0.163 − 0.282i)13-s + (0.0192 − 0.0333i)14-s + (−0.470 − 0.814i)16-s + 1.06·17-s + 1.14·19-s + (0.243 + 0.422i)20-s + (0.0373 − 0.0647i)22-s + (0.754 − 1.30i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $0.806 + 0.591i$
Analytic conductor: \(6.25228\)
Root analytic conductor: \(2.50045\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 783,\ (\ :1/2),\ 0.806 + 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61145 - 0.527401i\)
\(L(\frac12)\) \(\approx\) \(1.61145 - 0.527401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.100 + 0.173i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.556 - 0.964i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.359 - 0.622i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.874 - 1.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.587 + 1.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.38T + 17T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 + (-3.61 + 6.26i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (0.127 - 0.220i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.12T + 37T^{2} \)
41 \( 1 + (-2.52 + 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.94 + 6.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.05 - 3.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.37T + 53T^{2} \)
59 \( 1 + (1.94 - 3.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.59 - 4.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.74 + 6.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.68T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + (-6.19 - 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.47 - 9.47i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.231T + 89T^{2} \)
97 \( 1 + (-5.16 - 8.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.34919322558312371652054855910, −9.506070297028001909322267329278, −8.625261487426598115977263467933, −7.37515453670735220183598908516, −6.86522712909953740828674386386, −5.70239856477645631549188708032, −5.04721893846865060829551843462, −3.55017496591453477287074930882, −2.47359665137835809859434733088, −1.08494816576396977923327358120, 1.30304353423381473875338654881, 3.02116468375862329766293713904, 3.75883505575968947504122775195, 4.99602267381496048722148033630, 6.05023366969664551188774164353, 7.16291339630465789296139291282, 7.73079874607738205471591390122, 8.596378465669394032727617838831, 9.357620769967471496769132276054, 10.42878087336872256713471205929

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