↑

Properties

Label 2-42e2-63.5-c1-0-31
Degree $2$
Conductor $1764$
Sign $0.385 + 0.922i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (0.981 + 1.42i)3-s + (−0.276 − 0.479i)5-s + (−1.07 + 2.80i)9-s + (−4.03 − 2.32i)11-s + (−3.58 − 2.06i)13-s + (0.412 − 0.866i)15-s + (−3.62 − 6.27i)17-s + (5.81 + 3.35i)19-s + (4.85 − 2.80i)23-s + (2.34 − 4.06i)25-s + (−5.05 + 1.22i)27-s + (1.16 − 0.673i)29-s − 0.959i·31-s + (−0.637 − 8.04i)33-s + (3.53 − 6.12i)37-s + ⋯
L(s)  = 1  + (0.566 + 0.823i)3-s + (−0.123 − 0.214i)5-s + (−0.357 + 0.933i)9-s + (−1.21 − 0.702i)11-s + (−0.993 − 0.573i)13-s + (0.106 − 0.223i)15-s + (−0.878 − 1.52i)17-s + (1.33 + 0.770i)19-s + (1.01 − 0.584i)23-s + (0.469 − 0.812i)25-s + (−0.972 + 0.234i)27-s + (0.216 − 0.125i)29-s − 0.172i·31-s + (−0.110 − 1.40i)33-s + (0.581 − 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.385 + 0.922i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254173322\)
\(L(\frac12)\) \(\approx\) \(1.254173322\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.981 - 1.42i)T \)
7 \( 1 \)
good5 \( 1 + (0.276 + 0.479i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.03 + 2.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.58 + 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.62 + 6.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.81 - 3.35i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.85 + 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.16 + 0.673i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.959iT - 31T^{2} \)
37 \( 1 + (-3.53 + 6.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.02 + 1.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.80T + 47T^{2} \)
53 \( 1 + (-7.30 + 4.21i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.79T + 59T^{2} \)
61 \( 1 - 6.20iT - 61T^{2} \)
67 \( 1 + 3.37T + 67T^{2} \)
71 \( 1 - 0.407iT - 71T^{2} \)
73 \( 1 + (7.47 - 4.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.636T + 79T^{2} \)
83 \( 1 + (2.78 + 4.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.46 - 6.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 - 4.32i)T + (48.5 - 84.0i)T^{2} \)
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

−9.182110532541281780007922533136, −8.407692164633015369191131510923, −7.72835957752854645840198212310, −6.99912981801655016424292412219, −5.44416071811655741910846429506, −5.15626734731864315634662510684, −4.20276899461222091693224950452, −2.91807942694808013447255230465, −2.58849914845349230522097374080, −0.42689994950759106385410891366, 1.42761042218002406715430008486, 2.51530885075662230211169353998, 3.22014773286168319452697215545, 4.55283139884286502102815091248, 5.34987123619096877288357747001, 6.55768622207009043821858845272, 7.17090491608200159425062764754, 7.73007775935477516184129552415, 8.565962312977049906543056999004, 9.394399190838020760824197935970

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