↑

Properties

Label 2-770-35.27-c1-0-6
Degree $2$
Conductor $770$
Sign $0.604 - 0.796i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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  + (−0.707 − 0.707i)2-s + (0.0715 + 0.0715i)3-s + 1.00i·4-s + (−1.95 − 1.08i)5-s − 0.101i·6-s + (0.240 + 2.63i)7-s + (0.707 − 0.707i)8-s − 2.98i·9-s + (0.610 + 2.15i)10-s − 11-s + (−0.0715 + 0.0715i)12-s + (2.32 + 2.32i)13-s + (1.69 − 2.03i)14-s + (−0.0617 − 0.217i)15-s − 1.00·16-s + (−2.41 + 2.41i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.0412 + 0.0412i)3-s + 0.500i·4-s + (−0.873 − 0.487i)5-s − 0.0412i·6-s + (0.0908 + 0.995i)7-s + (0.250 − 0.250i)8-s − 0.996i·9-s + (0.193 + 0.680i)10-s − 0.301·11-s + (−0.0206 + 0.0206i)12-s + (0.645 + 0.645i)13-s + (0.452 − 0.543i)14-s + (−0.0159 − 0.0561i)15-s − 0.250·16-s + (−0.585 + 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671720 + 0.333298i\)
\(L(\frac12)\) \(\approx\) \(0.671720 + 0.333298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.95 + 1.08i)T \)
7 \( 1 + (-0.240 - 2.63i)T \)
11 \( 1 + T \)
good3 \( 1 + (-0.0715 - 0.0715i)T + 3iT^{2} \)
13 \( 1 + (-2.32 - 2.32i)T + 13iT^{2} \)
17 \( 1 + (2.41 - 2.41i)T - 17iT^{2} \)
19 \( 1 + 2.20T + 19T^{2} \)
23 \( 1 + (-1.77 + 1.77i)T - 23iT^{2} \)
29 \( 1 - 5.97iT - 29T^{2} \)
31 \( 1 - 4.99iT - 31T^{2} \)
37 \( 1 + (-6.33 - 6.33i)T + 37iT^{2} \)
41 \( 1 - 8.47iT - 41T^{2} \)
43 \( 1 + (1.16 - 1.16i)T - 43iT^{2} \)
47 \( 1 + (-4.80 + 4.80i)T - 47iT^{2} \)
53 \( 1 + (-3.59 + 3.59i)T - 53iT^{2} \)
59 \( 1 + 3.32T + 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + (-9.53 - 9.53i)T + 67iT^{2} \)
71 \( 1 + 8.78T + 71T^{2} \)
73 \( 1 + (-2.07 - 2.07i)T + 73iT^{2} \)
79 \( 1 + 2.87iT - 79T^{2} \)
83 \( 1 + (3.19 + 3.19i)T + 83iT^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + (-3.29 + 3.29i)T - 97iT^{2} \)
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

−10.50648380005341035550341258383, −9.379498503787004408771429259745, −8.639090052187658200678757645942, −8.407095275907195829805354631833, −7.03722630965946838500429055314, −6.17710731341849898081376153290, −4.82923423823984321880873032126, −3.87946763698703790504456301712, −2.83881413913508863925902825918, −1.31598951751629948796576071438, 0.49056046769443886222466668378, 2.39521343910531986753767248650, 3.83814591097902482176440848879, 4.72991485034146718635247032341, 5.94064832251749961097066108443, 7.01333320009466314253130963812, 7.72334246598465472910410159797, 8.107804041160081739441541993956, 9.266199461030494520498105749495, 10.38439102161390765467967525283

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