Properties

Label 2-770-385.139-c1-0-8
Degree $2$
Conductor $770$
Sign $-0.111 - 0.993i$
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 + (−0.463 + 1.42i)3-s + (0.309 + 0.951i)4-s + (−2.23 − 0.0266i)5-s + (1.21 − 0.881i)6-s + (−1.05 − 2.42i)7-s + (0.309 − 0.951i)8-s + (0.606 + 0.440i)9-s + (1.79 + 1.33i)10-s + (3.30 − 0.327i)11-s − 1.50·12-s + (0.191 − 0.263i)13-s + (−0.576 + 2.58i)14-s + (1.07 − 3.17i)15-s + (−0.809 + 0.587i)16-s + (0.316 + 0.435i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.267 + 0.823i)3-s + (0.154 + 0.475i)4-s + (−0.999 − 0.0119i)5-s + (0.495 − 0.359i)6-s + (−0.397 − 0.917i)7-s + (0.109 − 0.336i)8-s + (0.202 + 0.146i)9-s + (0.567 + 0.422i)10-s + (0.995 − 0.0987i)11-s − 0.433·12-s + (0.0530 − 0.0730i)13-s + (−0.154 + 0.690i)14-s + (0.277 − 0.820i)15-s + (−0.202 + 0.146i)16-s + (0.0767 + 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.111 - 0.993i$
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: \((2,\ 770,\ (\ :1/2),\ -0.111 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413811 + 0.462991i\)
\(L(\frac12)\) \(\approx\) \(0.413811 + 0.462991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0266i)T \)
7 \( 1 + (1.05 + 2.42i)T \)
11 \( 1 + (-3.30 + 0.327i)T \)
good3 \( 1 + (0.463 - 1.42i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-0.191 + 0.263i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.316 - 0.435i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.758 - 2.33i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.70iT - 23T^{2} \)
29 \( 1 + (2.28 - 0.741i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.248 - 0.341i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.07 - 0.999i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.25 - 3.86i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.787T + 43T^{2} \)
47 \( 1 + (1.98 - 6.10i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.90 - 8.13i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.34 - 1.08i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.60 - 1.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.10iT - 67T^{2} \)
71 \( 1 + (4.55 - 3.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-13.4 + 4.38i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.136 + 0.187i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.56 - 9.03i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.96iT - 89T^{2} \)
97 \( 1 + (-9.86 - 7.16i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.64253281087636819314113307579, −9.733216357841880880118998008739, −9.127662641453925600903687063397, −7.945511407952561465416103952553, −7.33931519051996409530159012521, −6.32099268292663665584547085761, −4.80380475203793299798371173180, −3.91493367517018164872183283675, −3.40546752952083331170861976275, −1.30933867795122804112873383399, 0.44060868694972057473810372028, 1.97482117214238129345245892636, 3.51514793916860587752317479940, 4.76604221060869849097290220184, 6.07740678713866013591941861006, 6.73520367794808854736267437077, 7.36391369116899787714874304713, 8.406568684528202620251363163459, 9.005129675963577066103693070542, 9.903644385618407798221975572432

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