Properties

Label 2-770-11.5-c1-0-16
Degree $2$
Conductor $770$
Sign $-0.874 + 0.484i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.929 − 0.675i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.355 + 1.09i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.519 − 1.59i)9-s − 0.999·10-s + (3.22 − 0.780i)11-s + 1.14·12-s + (1.17 + 3.61i)13-s + (−0.809 − 0.587i)14-s + (−0.929 + 0.675i)15-s + (0.309 − 0.951i)16-s + (2.00 − 6.16i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.536 − 0.389i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.144 + 0.446i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (−0.173 − 0.532i)9-s − 0.316·10-s + (0.971 − 0.235i)11-s + 0.331·12-s + (0.325 + 1.00i)13-s + (−0.216 − 0.157i)14-s + (−0.240 + 0.174i)15-s + (0.0772 − 0.237i)16-s + (0.485 − 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.874 + 0.484i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259488 - 1.00365i\)
\(L(\frac12)\) \(\approx\) \(0.259488 - 1.00365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.22 + 0.780i)T \)
good3 \( 1 + (0.929 + 0.675i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-1.17 - 3.61i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.00 + 6.16i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.16 + 1.57i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.65T + 23T^{2} \)
29 \( 1 + (0.313 - 0.227i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.85 + 5.72i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.93 - 1.40i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.66 + 5.57i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.66T + 43T^{2} \)
47 \( 1 + (-3.62 - 2.63i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.57 + 4.84i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.83 - 2.78i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.60 + 11.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.71T + 67T^{2} \)
71 \( 1 + (-1.02 + 3.16i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.1 - 8.82i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.86 + 5.74i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.44 - 10.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.87T + 89T^{2} \)
97 \( 1 + (1.02 + 3.14i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.856274759672038027039409084524, −9.170843470656115353544877056221, −8.574793400910922314519733376648, −7.23463440646910579950834670324, −6.55575625592699210610243668337, −5.41527078363167880647961864916, −4.40266959609217051734901214528, −3.35882045884041662450073943445, −1.77954600747846789035589353804, −0.65052899070501946916026009885, 1.62150321867411606631425156849, 3.40556581968031769065451451346, 4.54920242598209250854149289369, 5.57133720808075163093377264814, 6.14616740412755423089659072184, 7.15572454411531309058152591396, 8.203066021251277770394225232574, 8.746623395791444314674062260616, 10.05274164971746665404574720427, 10.47613378870921301943990773847

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