↑

Properties

Label 2-700-140.19-c1-0-21
Degree $2$
Conductor $700$
Sign $0.959 + 0.280i$
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  + (0.242 − 1.39i)2-s + (−0.703 + 0.406i)3-s + (−1.88 − 0.675i)4-s + (0.395 + 1.07i)6-s + (−2.62 + 0.336i)7-s + (−1.39 + 2.45i)8-s + (−1.17 + 2.02i)9-s + (4.20 − 2.43i)11-s + (1.59 − 0.289i)12-s + 0.895·13-s + (−0.167 + 3.73i)14-s + (3.08 + 2.54i)16-s + (2.94 + 5.10i)17-s + (2.54 + 2.12i)18-s + (1.45 − 2.52i)19-s + ⋯
L(s)  = 1  + (0.171 − 0.985i)2-s + (−0.406 + 0.234i)3-s + (−0.941 − 0.337i)4-s + (0.161 + 0.440i)6-s + (−0.991 + 0.127i)7-s + (−0.494 + 0.869i)8-s + (−0.390 + 0.675i)9-s + (1.26 − 0.732i)11-s + (0.461 − 0.0835i)12-s + 0.248·13-s + (−0.0447 + 0.998i)14-s + (0.771 + 0.635i)16-s + (0.714 + 1.23i)17-s + (0.598 + 0.500i)18-s + (0.334 − 0.579i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10468 - 0.157820i\)
\(L(\frac12)\) \(\approx\) \(1.10468 - 0.157820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.242 + 1.39i)T \)
5 \( 1 \)
7 \( 1 + (2.62 - 0.336i)T \)
good3 \( 1 + (0.703 - 0.406i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-4.20 + 2.43i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.895T + 13T^{2} \)
17 \( 1 + (-2.94 - 5.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 2.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.780 - 1.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.73T + 29T^{2} \)
31 \( 1 + (-2.20 - 3.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.57 - 0.910i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.4iT - 41T^{2} \)
43 \( 1 + 3.04T + 43T^{2} \)
47 \( 1 + (4.52 + 2.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.155 - 0.0898i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.68 - 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.33 + 3.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.89 - 6.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.38iT - 71T^{2} \)
73 \( 1 + (-4.39 - 7.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.14 + 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.03iT - 83T^{2} \)
89 \( 1 + (1.52 + 0.882i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.83T + 97T^{2} \)
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

−10.37732858714273105491763768109, −9.877271943026560985217617420470, −8.830051282712496892766351550347, −8.212516305641829087916399569653, −6.48679166083945036150125961143, −5.88991801779595012663048193549, −4.77439498770825449457264918751, −3.68275754525775056220681646575, −2.85068034184047666445874596296, −1.15989059878720317015764172330, 0.74065598959010523425400645454, 3.16950284803098353564298347988, 4.08974421654726649942623599203, 5.28631345279149100881896020119, 6.35072666173808299793520383789, 6.67248283957904026226297151201, 7.61540368270825934740623384733, 8.804463928619651134317679501195, 9.511149837365768891658254302243, 10.10479681712294880409898446964

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