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Properties

Label 2-783-9.4-c1-0-24
Degree $2$
Conductor $783$
Sign $-0.626 - 0.779i$
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  + (−0.788 − 1.36i)2-s + (−0.244 + 0.423i)4-s + (−0.409 + 0.709i)5-s + (−1.13 − 1.96i)7-s − 2.38·8-s + 1.29·10-s + (0.608 + 1.05i)11-s + (3.16 − 5.49i)13-s + (−1.79 + 3.10i)14-s + (2.36 + 4.10i)16-s − 5.28·17-s − 4.32·19-s + (−0.200 − 0.346i)20-s + (0.960 − 1.66i)22-s + (1.77 − 3.07i)23-s + ⋯
L(s)  = 1  + (−0.557 − 0.966i)2-s + (−0.122 + 0.211i)4-s + (−0.183 + 0.317i)5-s + (−0.429 − 0.744i)7-s − 0.842·8-s + 0.408·10-s + (0.183 + 0.317i)11-s + (0.879 − 1.52i)13-s + (−0.479 + 0.830i)14-s + (0.592 + 1.02i)16-s − 1.28·17-s − 0.991·19-s + (−0.0447 − 0.0775i)20-s + (0.204 − 0.354i)22-s + (0.370 − 0.641i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $-0.626 - 0.779i$
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: \((2,\ 783,\ (\ :1/2),\ -0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153319 + 0.319953i\)
\(L(\frac12)\) \(\approx\) \(0.153319 + 0.319953i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.788 + 1.36i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.409 - 0.709i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.13 + 1.96i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.608 - 1.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.16 + 5.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (-1.77 + 3.07i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (4.46 - 7.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + (2.70 - 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.26 + 2.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.91 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.56T + 53T^{2} \)
59 \( 1 + (-0.640 + 1.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.816 - 1.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.70 - 8.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 5.25T + 73T^{2} \)
79 \( 1 + (1.98 + 3.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.27 - 3.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.69T + 89T^{2} \)
97 \( 1 + (-6.35 - 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15050991186521958724772055958, −8.900759468269599406170980808634, −8.458856191028325268509559517149, −7.04053484663496699212780374981, −6.46161295592975265669331557976, −5.19723675328864455992878018241, −3.76270537986043875377223049634, −3.04320493002936493138627417601, −1.67530460561364003391975085965, −0.20429849400487563534855427118, 2.06520675649407066605153687031, 3.52347145647478354998631201274, 4.66223701873775969735395701214, 6.06188204836353769840978830375, 6.43919778692201284126797903756, 7.34087820296437425770686949856, 8.569447140099549165284603690545, 8.851403309863604159231360470359, 9.460042429802095574362642802030, 10.90473512988764949519128796126

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