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Properties

Label 2-770-385.139-c1-0-47
Degree $2$
Conductor $770$
Sign $-0.976 + 0.215i$
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  + (0.809 + 0.587i)2-s + (0.934 − 2.87i)3-s + (0.309 + 0.951i)4-s + (−2.04 − 0.902i)5-s + (2.44 − 1.77i)6-s + (−2.25 + 1.38i)7-s + (−0.309 + 0.951i)8-s + (−4.97 − 3.61i)9-s + (−1.12 − 1.93i)10-s + (−2.96 + 1.48i)11-s + 3.02·12-s + (0.321 − 0.442i)13-s + (−2.63 − 0.203i)14-s + (−4.50 + 5.03i)15-s + (−0.809 + 0.587i)16-s + (−3.59 − 4.94i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.539 − 1.66i)3-s + (0.154 + 0.475i)4-s + (−0.914 − 0.403i)5-s + (0.998 − 0.725i)6-s + (−0.851 + 0.523i)7-s + (−0.109 + 0.336i)8-s + (−1.65 − 1.20i)9-s + (−0.355 − 0.611i)10-s + (−0.893 + 0.448i)11-s + 0.872·12-s + (0.0892 − 0.122i)13-s + (−0.705 − 0.0543i)14-s + (−1.16 + 1.30i)15-s + (−0.202 + 0.146i)16-s + (−0.871 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.976 + 0.215i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0973024 - 0.892417i\)
\(L(\frac12)\) \(\approx\) \(0.0973024 - 0.892417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (2.04 + 0.902i)T \)
7 \( 1 + (2.25 - 1.38i)T \)
11 \( 1 + (2.96 - 1.48i)T \)
good3 \( 1 + (-0.934 + 2.87i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-0.321 + 0.442i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.59 + 4.94i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0567 - 0.174i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.00iT - 23T^{2} \)
29 \( 1 + (4.80 - 1.55i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.63 + 3.63i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.53 + 1.47i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.44 - 10.5i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.20T + 43T^{2} \)
47 \( 1 + (0.0177 - 0.0547i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.85 + 10.8i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.59 + 1.81i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.98 - 2.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.61iT - 67T^{2} \)
71 \( 1 + (-0.266 + 0.193i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.98 + 2.26i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.23 - 3.08i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.14 + 4.32i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.52iT - 89T^{2} \)
97 \( 1 + (8.96 + 6.51i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.642425680537710434007953232051, −8.549654571830401815960870593666, −8.143476384486793872027140531847, −7.11900243252931435298704804837, −6.75542071961447633496921785331, −5.63903184357534221436918573257, −4.47821095301114555620425501029, −3.04629480479468296240362171627, −2.35733231061709293098604669373, −0.32524614698608542847750256761, 2.64665080878189238340950639340, 3.67616699831219940829144597116, 3.87586312361298632823139838094, 5.01487217773003551162348018084, 6.05329675872563360487400306896, 7.32884129180587970452549196735, 8.351950513819457875818875822441, 9.229535754500437821057774675201, 10.09923184724810286169726222285, 10.74327480373486857712812768298

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