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Properties

Label 2-770-11.5-c1-0-9
Degree $2$
Conductor $770$
Sign $-0.570 - 0.821i$
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 + (2.13 + 1.55i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.817 + 2.51i)6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (1.23 + 3.79i)9-s + 0.999·10-s + (1.10 + 3.12i)11-s − 2.64·12-s + (0.945 + 2.91i)13-s + (−0.809 − 0.587i)14-s + (2.13 − 1.55i)15-s + (0.309 − 0.951i)16-s + (−0.0211 + 0.0650i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.23 + 0.897i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.333 + 1.02i)6-s + (−0.305 + 0.222i)7-s + (−0.286 − 0.207i)8-s + (0.411 + 1.26i)9-s + 0.316·10-s + (0.331 + 0.943i)11-s − 0.763·12-s + (0.262 + 0.807i)13-s + (−0.216 − 0.157i)14-s + (0.552 − 0.401i)15-s + (0.0772 − 0.237i)16-s + (−0.00512 + 0.0157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.570 - 0.821i$
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: \((2,\ 770,\ (\ :1/2),\ -0.570 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11687 + 2.13479i\)
\(L(\frac12)\) \(\approx\) \(1.11687 + 2.13479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 - 3.12i)T \)
good3 \( 1 + (-2.13 - 1.55i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-0.945 - 2.91i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.0211 - 0.0650i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.24 + 1.62i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 + (2.67 - 1.94i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.27 - 3.92i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.55 + 3.31i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.15 - 3.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.84T + 43T^{2} \)
47 \( 1 + (9.64 + 7.00i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0382 - 0.117i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.05 + 6.57i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.783 + 2.41i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.20T + 67T^{2} \)
71 \( 1 + (-4.24 + 13.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.65 - 4.10i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.83 + 11.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.943 - 2.90i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (0.848 + 2.61i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.19435827846284223313768192098, −9.411743881686768003780953994667, −9.003135047772822515729368186930, −8.248618313342964861079221587234, −7.21782344400353387611568078374, −6.30158909091388723327970199567, −4.95695170661795116817225434684, −4.30122075163459784275669608491, −3.38258748699489829765263283484, −2.08998242701261265997254080841, 1.06542506420723677777784721069, 2.42955165111583841256077787489, 3.16872426489937660281620128520, 4.03341077438659760012393465452, 5.72251916041346612009771002764, 6.55110232768070903402060546321, 7.64611399763742029165075581413, 8.325448163333762605636549383382, 9.134451885417076822320468258189, 9.990016338504426560077051025470

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