↑

Properties

Label 2-700-140.123-c1-0-46
Degree $2$
Conductor $700$
Sign $-0.961 + 0.273i$
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  + (−0.157 − 1.40i)2-s + (−0.290 − 1.08i)3-s + (−1.95 + 0.442i)4-s + (−1.47 + 0.578i)6-s + (2.16 − 1.51i)7-s + (0.928 + 2.67i)8-s + (1.50 − 0.871i)9-s + (2.58 + 1.49i)11-s + (1.04 + 1.98i)12-s + (−4.05 − 4.05i)13-s + (−2.47 − 2.80i)14-s + (3.60 − 1.72i)16-s + (−0.617 − 2.30i)17-s + (−1.46 − 1.98i)18-s + (−1.25 − 2.17i)19-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)2-s + (−0.167 − 0.625i)3-s + (−0.975 + 0.221i)4-s + (−0.602 + 0.236i)6-s + (0.819 − 0.573i)7-s + (0.328 + 0.944i)8-s + (0.503 − 0.290i)9-s + (0.779 + 0.449i)11-s + (0.301 + 0.572i)12-s + (−1.12 − 1.12i)13-s + (−0.661 − 0.750i)14-s + (0.902 − 0.431i)16-s + (−0.149 − 0.558i)17-s + (−0.344 − 0.467i)18-s + (−0.288 − 0.499i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.961 + 0.273i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174152 - 1.25032i\)
\(L(\frac12)\) \(\approx\) \(0.174152 - 1.25032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.157 + 1.40i)T \)
5 \( 1 \)
7 \( 1 + (-2.16 + 1.51i)T \)
good3 \( 1 + (0.290 + 1.08i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.58 - 1.49i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.05 + 4.05i)T + 13iT^{2} \)
17 \( 1 + (0.617 + 2.30i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.25 + 2.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.79 - 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + (-3.12 - 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.61 + 1.23i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 + (7.62 - 7.62i)T - 43iT^{2} \)
47 \( 1 + (0.765 - 2.85i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.47 + 0.662i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (3.00 - 0.805i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.09iT - 71T^{2} \)
73 \( 1 + (-9.41 + 2.52i)T + (63.2 - 36.5i)T^{2} \)
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

−10.09548340597303719396496471995, −9.464885983974448971852075931277, −8.350174682247813817031081606871, −7.48419537219090819885138136719, −6.77696889001610513439815917583, −5.13162620025058669295453411247, −4.51466277516266671255832107259, −3.21861602371553736939239960317, −1.88327543983330962833643942730, −0.76723114888196137425169254843, 1.74253506588032178198049134842, 3.78974569419935376623952959772, 4.73116545066479670864137253212, 5.27444081829205892813151264583, 6.53749334341102533271697435273, 7.21952487396035753367341079991, 8.396067737757433708168156003063, 8.946681743489789279938254275585, 9.825192573419066842970435202053, 10.61154133194336964630273747061

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