↑

Properties

Label 2-644-7.4-c1-0-13
Degree $2$
Conductor $644$
Sign $-0.999 - 0.0347i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + (1.04 − 1.80i)3-s + (−1.94 − 3.36i)5-s + (−2.32 − 1.25i)7-s + (−0.673 − 1.16i)9-s + (−2.28 + 3.95i)11-s + 2.58·13-s − 8.09·15-s + (2.22 − 3.85i)17-s + (−1.93 − 3.35i)19-s + (−4.69 + 2.89i)21-s + (0.5 + 0.866i)23-s + (−5.03 + 8.72i)25-s + 3.44·27-s − 6.62·29-s + (−2.51 + 4.35i)31-s + ⋯
L(s)  = 1  + (0.601 − 1.04i)3-s + (−0.868 − 1.50i)5-s + (−0.879 − 0.474i)7-s + (−0.224 − 0.388i)9-s + (−0.689 + 1.19i)11-s + 0.715·13-s − 2.08·15-s + (0.539 − 0.934i)17-s + (−0.444 − 0.769i)19-s + (−1.02 + 0.631i)21-s + (0.104 + 0.180i)23-s + (−1.00 + 1.74i)25-s + 0.663·27-s − 1.22·29-s + (−0.451 + 0.782i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.999 - 0.0347i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ -0.999 - 0.0347i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0171387 + 0.986963i\)
\(L(\frac12)\) \(\approx\) \(0.0171387 + 0.986963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2.32 + 1.25i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-1.04 + 1.80i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.94 + 3.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.28 - 3.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + (-2.22 + 3.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.93 + 3.35i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 6.62T + 29T^{2} \)
31 \( 1 + (2.51 - 4.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.14 + 5.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.21T + 41T^{2} \)
43 \( 1 - 7.89T + 43T^{2} \)
47 \( 1 + (4.74 + 8.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.34 + 9.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.915 - 1.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.14 + 1.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.962 + 1.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (-0.660 + 1.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.22 - 3.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.87T + 83T^{2} \)
89 \( 1 + (5.50 + 9.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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

−9.911369624351252314072886459475, −9.027180830145580461414666234084, −8.355537898783493778477174852413, −7.32222502518330627216773334558, −7.12152867259953235409859345006, −5.45214923430883647056673279276, −4.49091271812548644220386782301, −3.37372366949960680383849434441, −1.87375859532445318252197567140, −0.48757578626258412398192126791, 2.74430575828087890733374398466, 3.49264931778349468242649102637, 3.93414633684708488031511516722, 5.77115782937933373511339802413, 6.41693738407143874568853812248, 7.68758977984599008411013293758, 8.416254322169747681452041106558, 9.323478519090598843387111893984, 10.39989831832678263426816922969, 10.65639704291819755447117535380

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