↑

Properties

Label 2-700-140.67-c1-0-27
Degree $2$
Conductor $700$
Sign $0.738 - 0.674i$
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  + (1.40 − 0.157i)2-s + (−1.08 + 0.290i)3-s + (1.95 − 0.442i)4-s + (−1.47 + 0.578i)6-s + (1.51 + 2.16i)7-s + (2.67 − 0.928i)8-s + (−1.50 + 0.871i)9-s + (2.58 + 1.49i)11-s + (−1.98 + 1.04i)12-s + (−4.05 + 4.05i)13-s + (2.47 + 2.80i)14-s + (3.60 − 1.72i)16-s + (2.30 − 0.617i)17-s + (−1.98 + 1.46i)18-s + (1.25 + 2.17i)19-s + ⋯
L(s)  = 1  + (0.993 − 0.111i)2-s + (−0.625 + 0.167i)3-s + (0.975 − 0.221i)4-s + (−0.602 + 0.236i)6-s + (0.573 + 0.819i)7-s + (0.944 − 0.328i)8-s + (−0.503 + 0.290i)9-s + (0.779 + 0.449i)11-s + (−0.572 + 0.301i)12-s + (−1.12 + 1.12i)13-s + (0.661 + 0.750i)14-s + (0.902 − 0.431i)16-s + (0.558 − 0.149i)17-s + (−0.467 + 0.344i)18-s + (0.288 + 0.499i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.738 - 0.674i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22826 + 0.864824i\)
\(L(\frac12)\) \(\approx\) \(2.22826 + 0.864824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.157i)T \)
5 \( 1 \)
7 \( 1 + (-1.51 - 2.16i)T \)
good3 \( 1 + (1.08 - 0.290i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.58 - 1.49i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.05 - 4.05i)T - 13iT^{2} \)
17 \( 1 + (-2.30 + 0.617i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.25 - 2.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.55 + 5.79i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 + (-3.12 - 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.23 + 4.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 + (-7.62 - 7.62i)T + 43iT^{2} \)
47 \( 1 + (2.85 + 0.765i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.662 + 2.47i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.04 - 1.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.950 - 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.805 + 3.00i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.09iT - 71T^{2} \)
73 \( 1 + (2.52 + 9.41i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.03 + 6.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.99 + 5.99i)T + 83iT^{2} \)
89 \( 1 + (-1.77 + 1.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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

−10.83371085377591587776381417511, −9.906232146422465180807162532451, −8.896760478631694918213620853074, −7.72672310201059831092264350389, −6.72256230753446544346427685877, −5.91867672343282019929311909657, −4.95331272811914019216028396562, −4.46094081884682910586210530317, −2.89117470854133765283626770559, −1.79933981070959694734861749144, 1.07828442414922392077422655842, 2.87425969082189828678806330034, 3.88304308254501665560073766459, 5.08168857727926625541413976479, 5.64029497070321049206613541344, 6.70994511086825033067996852741, 7.46088812761779997401099474833, 8.309487263671502879530880024102, 9.759267771151396241961186118119, 10.63603412089440317806872502917

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