↑

Properties

Label 2-783-29.23-c1-0-0
Degree $2$
Conductor $783$
Sign $0.270 + 0.962i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 + 1.40i)2-s + (−0.277 − 1.21i)4-s + (−1.27 + 1.60i)5-s + (−0.599 + 2.62i)7-s + (−1.22 − 0.588i)8-s + (−0.821 − 3.59i)10-s + (−3.82 + 1.84i)11-s + (−0.0990 + 0.0476i)13-s + (−3.02 − 3.79i)14-s + (4.44 − 2.14i)16-s + 1.10·17-s + (−0.291 − 1.27i)19-s + (2.30 + 1.10i)20-s + (1.70 − 7.46i)22-s + (−1.74 − 2.19i)23-s + ⋯
L(s)  = 1  + (−0.794 + 0.996i)2-s + (−0.138 − 0.607i)4-s + (−0.571 + 0.716i)5-s + (−0.226 + 0.991i)7-s + (−0.432 − 0.208i)8-s + (−0.259 − 1.13i)10-s + (−1.15 + 0.555i)11-s + (−0.0274 + 0.0132i)13-s + (−0.808 − 1.01i)14-s + (1.11 − 0.535i)16-s + 0.269·17-s + (−0.0667 − 0.292i)19-s + (0.514 + 0.247i)20-s + (0.363 − 1.59i)22-s + (−0.364 − 0.456i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(6.25228\)
Root analytic conductor: \(2.50045\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 783,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120846 - 0.0915952i\)
\(L(\frac12)\) \(\approx\) \(0.120846 - 0.0915952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.97 - 4.48i)T \)
good2 \( 1 + (1.12 - 1.40i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (1.27 - 1.60i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.599 - 2.62i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (3.82 - 1.84i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.0990 - 0.0476i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 1.10T + 17T^{2} \)
19 \( 1 + (0.291 + 1.27i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (1.74 + 2.19i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-2.77 + 3.48i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (8.32 + 4.00i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 6.19T + 41T^{2} \)
43 \( 1 + (2.28 + 2.86i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-6.49 + 3.12i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-7.09 + 8.90i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 + (2.02 - 8.86i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (1.96 + 0.948i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-4.57 + 2.20i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (5.57 + 6.99i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (7.70 + 3.71i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (0.900 + 3.94i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (6.64 - 8.33i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-2.53 - 11.1i)T + (-87.3 + 42.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.62272370743619102764596924534, −10.04011534362419660475974657673, −8.954013447089394817277965774414, −8.391776448016823916153426032293, −7.45152442807206075479101394264, −6.95158795704247035944906205938, −5.92980660864872619223864132841, −5.06222464688342987449959705107, −3.44789437724004994845068043411, −2.45533564198015254465635371955, 0.10676608236265841023268603049, 1.23671864804845127172890714645, 2.78411844196195788020739694965, 3.78578402956783684336031327905, 4.93277471260519742556623665527, 6.04401155095522824917022368219, 7.38641809491165792923965396779, 8.234653094930167949740130729815, 8.725653539922016905761860014999, 10.02788254635625454112815174751

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