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Properties

Label 2-783-87.26-c1-0-31
Degree $2$
Conductor $783$
Sign $-0.168 + 0.985i$
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  + (0.561 + 0.352i)2-s + (−0.676 − 1.40i)4-s + (−0.901 − 3.95i)5-s + (3.71 + 1.78i)7-s + (0.264 − 2.34i)8-s + (0.887 − 2.53i)10-s + (0.0605 + 0.537i)11-s + (1.03 + 0.829i)13-s + (1.45 + 2.31i)14-s + (−0.969 + 1.21i)16-s + (3.42 − 3.42i)17-s + (−6.17 − 2.16i)19-s + (−4.94 + 3.94i)20-s + (−0.155 + 0.323i)22-s + (−1.09 − 0.250i)23-s + ⋯
L(s)  = 1  + (0.397 + 0.249i)2-s + (−0.338 − 0.702i)4-s + (−0.403 − 1.76i)5-s + (1.40 + 0.676i)7-s + (0.0934 − 0.829i)8-s + (0.280 − 0.802i)10-s + (0.0182 + 0.162i)11-s + (0.288 + 0.230i)13-s + (0.388 + 0.618i)14-s + (−0.242 + 0.303i)16-s + (0.831 − 0.831i)17-s + (−1.41 − 0.495i)19-s + (−1.10 + 0.881i)20-s + (−0.0331 + 0.0688i)22-s + (−0.229 − 0.0523i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(6.25228\)
Root analytic conductor: \(2.50045\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 783,\ (\ :1/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10665 - 1.31181i\)
\(L(\frac12)\) \(\approx\) \(1.10665 - 1.31181i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.20 + 4.32i)T \)
good2 \( 1 + (-0.561 - 0.352i)T + (0.867 + 1.80i)T^{2} \)
5 \( 1 + (0.901 + 3.95i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-3.71 - 1.78i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-0.0605 - 0.537i)T + (-10.7 + 2.44i)T^{2} \)
13 \( 1 + (-1.03 - 0.829i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-3.42 + 3.42i)T - 17iT^{2} \)
19 \( 1 + (6.17 + 2.16i)T + (14.8 + 11.8i)T^{2} \)
23 \( 1 + (1.09 + 0.250i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (-2.17 + 3.46i)T + (-13.4 - 27.9i)T^{2} \)
37 \( 1 + (0.116 + 0.0131i)T + (36.0 + 8.23i)T^{2} \)
41 \( 1 + (1.85 + 1.85i)T + 41iT^{2} \)
43 \( 1 + (2.37 - 1.49i)T + (18.6 - 38.7i)T^{2} \)
47 \( 1 + (-4.54 + 0.512i)T + (45.8 - 10.4i)T^{2} \)
53 \( 1 + (13.8 - 3.16i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 5.24iT - 59T^{2} \)
61 \( 1 + (-2.86 - 8.18i)T + (-47.6 + 38.0i)T^{2} \)
67 \( 1 + (-11.0 + 8.83i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-1.86 + 2.33i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-7.33 - 11.6i)T + (-31.6 + 65.7i)T^{2} \)
79 \( 1 + (-0.573 + 5.08i)T + (-77.0 - 17.5i)T^{2} \)
83 \( 1 + (5.49 + 11.4i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-12.0 - 7.56i)T + (38.6 + 80.1i)T^{2} \)
97 \( 1 + (0.400 - 1.14i)T + (-75.8 - 60.4i)T^{2} \)
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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

−9.858991069186800079633650061336, −9.034485242055108496938230605948, −8.455054675796476797515609770600, −7.72109125388919631811978719941, −6.22848100506234223430752740678, −5.31038092018513726404014656356, −4.74103488565117507001953639402, −4.17646098269097791892227787426, −1.95261946065248898730730671407, −0.809273608381150542815312180714, 1.98143953101624641413373364588, 3.28403813663288060620095639634, 3.89204972090038344906919941707, 4.90124470974368715155848439211, 6.25601119147551146956627799494, 7.17539041069156126937145569051, 8.140620198223489203800351795322, 8.276600554080740240475546876097, 10.07303915183733734218770855266, 10.84917428677683474022745475167

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