↑

Properties

Label 2-700-28.3-c1-0-47
Degree $2$
Conductor $700$
Sign $0.996 - 0.0821i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (0.501 + 1.32i)2-s + (0.895 − 1.55i)3-s + (−1.49 + 1.32i)4-s + (2.49 + 0.405i)6-s + (0.644 − 2.56i)7-s + (−2.50 − 1.31i)8-s + (−0.103 − 0.179i)9-s + (3.66 + 2.11i)11-s + (0.717 + 3.50i)12-s − 2.98i·13-s + (3.71 − 0.434i)14-s + (0.480 − 3.97i)16-s + (1.92 + 1.10i)17-s + (0.184 − 0.226i)18-s + (−2.28 − 3.95i)19-s + ⋯
L(s)  = 1  + (0.354 + 0.934i)2-s + (0.516 − 0.895i)3-s + (−0.748 + 0.663i)4-s + (1.02 + 0.165i)6-s + (0.243 − 0.969i)7-s + (−0.885 − 0.464i)8-s + (−0.0344 − 0.0596i)9-s + (1.10 + 0.638i)11-s + (0.207 + 1.01i)12-s − 0.827i·13-s + (0.993 − 0.116i)14-s + (0.120 − 0.992i)16-s + (0.465 + 0.268i)17-s + (0.0435 − 0.0533i)18-s + (−0.523 − 0.907i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.996 - 0.0821i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.996 - 0.0821i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12288 + 0.0872949i\)
\(L(\frac12)\) \(\approx\) \(2.12288 + 0.0872949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.501 - 1.32i)T \)
5 \( 1 \)
7 \( 1 + (-0.644 + 2.56i)T \)
good3 \( 1 + (-0.895 + 1.55i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3.66 - 2.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.98iT - 13T^{2} \)
17 \( 1 + (-1.92 - 1.10i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.28 + 3.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.78 + 1.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + (-1.20 + 2.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.16 + 3.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.88iT - 41T^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 + (3.38 + 5.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.41 - 11.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.99 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0195 + 0.0113i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.38 + 2.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.07iT - 71T^{2} \)
73 \( 1 + (-2.88 - 1.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.14 - 1.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (14.4 - 8.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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

−10.31257041944142397859058887082, −9.362230510473369215018167816380, −8.287483932398258250211018982704, −7.79838941412977377794160563436, −6.89501959347543161258900324375, −6.44565097158546615312466984459, −4.95117665632687185184084917711, −4.15008657182431615689365663646, −2.88209644780421367917579606380, −1.14505573382865114000488085633, 1.55657833016569898114496574699, 2.93875391056247378422505244744, 3.79055084833694586101740780952, 4.62786508571742246746527241323, 5.68332248785223462161535932020, 6.65827654344685321124084259454, 8.613212008785825140754328955767, 8.743642790260261972329308608428, 9.691259108105234908746986589728, 10.31439175250312382751406875481

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