↑

Properties

Label 2-700-28.3-c1-0-8
Degree $2$
Conductor $700$
Sign $-0.0969 - 0.995i$
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  + (0.950 − 1.04i)2-s + (−1.36 + 2.37i)3-s + (−0.192 − 1.99i)4-s + (1.18 + 3.68i)6-s + (1.02 + 2.43i)7-s + (−2.26 − 1.69i)8-s + (−2.24 − 3.89i)9-s + (0.0868 + 0.0501i)11-s + (4.98 + 2.26i)12-s + 4.11i·13-s + (3.52 + 1.24i)14-s + (−3.92 + 0.766i)16-s + (−4.67 − 2.69i)17-s + (−6.20 − 1.34i)18-s + (3.72 + 6.45i)19-s + ⋯
L(s)  = 1  + (0.672 − 0.740i)2-s + (−0.790 + 1.36i)3-s + (−0.0962 − 0.995i)4-s + (0.482 + 1.50i)6-s + (0.387 + 0.922i)7-s + (−0.801 − 0.597i)8-s + (−0.748 − 1.29i)9-s + (0.0261 + 0.0151i)11-s + (1.43 + 0.654i)12-s + 1.14i·13-s + (0.942 + 0.333i)14-s + (−0.981 + 0.191i)16-s + (−1.13 − 0.654i)17-s + (−1.46 − 0.317i)18-s + (0.855 + 1.48i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.0969 - 0.995i$
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: \((2,\ 700,\ (\ :1/2),\ -0.0969 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811270 + 0.894141i\)
\(L(\frac12)\) \(\approx\) \(0.811270 + 0.894141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.950 + 1.04i)T \)
5 \( 1 \)
7 \( 1 + (-1.02 - 2.43i)T \)
good3 \( 1 + (1.36 - 2.37i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.0868 - 0.0501i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.11iT - 13T^{2} \)
17 \( 1 + (4.67 + 2.69i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.72 - 6.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.30 - 0.754i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + (2.72 - 4.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.519 - 0.899i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.99iT - 41T^{2} \)
43 \( 1 - 7.04iT - 43T^{2} \)
47 \( 1 + (2.22 + 3.84i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.07 + 5.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.26 + 7.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.84 + 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0950 - 0.0548i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.73iT - 71T^{2} \)
73 \( 1 + (-5.32 - 3.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.70 + 2.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.50T + 83T^{2} \)
89 \( 1 + (2.76 - 1.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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

−10.95057477313060141473214066888, −9.742576413434970861852669049876, −9.556138280763105917772238734618, −8.482883262230110009392059107004, −6.70796447740191925442648244022, −5.76117552866979449537865736392, −5.06052079323104621803178840429, −4.33870696625822239648276742408, −3.36117450607207104707781931773, −1.89470397072037751532996220992, 0.54860577962350823983963777068, 2.31903046930836527658058864789, 3.86877593435436889135833994892, 5.05389826187720704298976785231, 5.83109488012811207766227064326, 6.77728533368900416010621796161, 7.34738176023544351618629812148, 7.967470793569147299326821294605, 9.020718288602930838920965606422, 10.63483758247962166761105821286

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