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Properties

Label 2-770-11.9-c1-0-9
Degree $2$
Conductor $770$
Sign $0.0219 - 0.999i$
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 + (0.5 − 0.363i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.190 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 2.48i)9-s + 0.999·10-s + (0.809 + 3.21i)11-s − 0.618·12-s + (−0.236 + 0.726i)13-s + (0.809 − 0.587i)14-s + (−0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.118 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.288 − 0.209i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.0779 + 0.239i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.829i)9-s + 0.316·10-s + (0.243 + 0.969i)11-s − 0.178·12-s + (−0.0654 + 0.201i)13-s + (0.216 − 0.157i)14-s + (−0.129 − 0.0937i)15-s + (0.0772 + 0.237i)16-s + (0.0286 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877352 + 0.858272i\)
\(L(\frac12)\) \(\approx\) \(0.877352 + 0.858272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 3.21i)T \)
good3 \( 1 + (-0.5 + 0.363i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (0.236 - 0.726i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.118 - 0.363i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.927 - 0.673i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2 - 6.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.47 - 3.97i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.11 + 2.26i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 + (7.47 - 5.42i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.23 - 6.88i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.78 + 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.0901 - 0.277i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + (2.14 + 6.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.11 - 2.26i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.23 + 3.80i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.663 + 2.04i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.14T + 89T^{2} \)
97 \( 1 + (3.28 - 10.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.43911081130940035145373782820, −9.377426053313712312314355309345, −8.833155331729983036498366610301, −7.82859159137928719969075310560, −7.21871121269412303424888371351, −6.32830580614028160642863279334, −5.07642476909370383560368844732, −4.46538170804989560808534817160, −2.94760118567541109248047283790, −1.42205800234605816658568412209, 0.71150018506487859814641346662, 2.64709525694873745018113978075, 3.31163370614067613175126902385, 4.30849656905041480001551140722, 5.71602392363153508871007039461, 6.57522159051783907433885961997, 7.70671430784639042364018710944, 8.718406093385376481556229902453, 9.254820599029903243090819771316, 10.05157434325184337418049454806

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