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Properties

Label 2-770-385.54-c1-0-13
Degree $2$
Conductor $770$
Sign $-0.816 - 0.577i$
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  + (−0.5 + 0.866i)2-s + (0.533 + 0.924i)3-s + (−0.499 − 0.866i)4-s + (1.13 + 1.92i)5-s − 1.06·6-s + (1.22 + 2.34i)7-s + 0.999·8-s + (0.930 − 1.61i)9-s + (−2.23 + 0.0212i)10-s + (−2.75 + 1.85i)11-s + (0.533 − 0.924i)12-s + 1.72i·13-s + (−2.64 − 0.116i)14-s + (−1.17 + 2.07i)15-s + (−0.5 + 0.866i)16-s + (−1.33 + 0.772i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.308 + 0.533i)3-s + (−0.249 − 0.433i)4-s + (0.508 + 0.861i)5-s − 0.435·6-s + (0.461 + 0.887i)7-s + 0.353·8-s + (0.310 − 0.537i)9-s + (−0.707 + 0.00673i)10-s + (−0.829 + 0.558i)11-s + (0.154 − 0.266i)12-s + 0.479i·13-s + (−0.706 − 0.0310i)14-s + (−0.303 + 0.536i)15-s + (−0.125 + 0.216i)16-s + (−0.324 + 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443499 + 1.39356i\)
\(L(\frac12)\) \(\approx\) \(0.443499 + 1.39356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-1.13 - 1.92i)T \)
7 \( 1 + (-1.22 - 2.34i)T \)
11 \( 1 + (2.75 - 1.85i)T \)
good3 \( 1 + (-0.533 - 0.924i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 1.72iT - 13T^{2} \)
17 \( 1 + (1.33 - 0.772i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.241 + 0.418i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.47 - 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.63iT - 29T^{2} \)
31 \( 1 + (-4.21 + 2.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.81 + 1.04i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.37T + 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + (-1.81 + 3.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.24 - 5.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.56 + 2.05i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.573 + 0.993i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.81 - 2.77i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 + (-6.69 + 3.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.64 - 4.99i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.34iT - 83T^{2} \)
89 \( 1 + (-6.49 - 3.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.99T + 97T^{2} \)
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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

−10.39910230225174433450243128065, −9.668192357021525458003350305879, −9.081134819412966340142921607239, −8.151224544874206408099872647362, −7.15255270155505600269844910313, −6.38897866717942728314021617431, −5.40941853961548798675959679948, −4.48376149664715241132803571520, −3.08221955202945798521214040390, −1.93686201317998150723399940381, 0.830216969593325251740130824635, 1.90710785363511210301800814881, 3.10851442394123760335599118548, 4.58688874018367141265189256535, 5.20546722208826897236221975401, 6.67463357257568837635774842678, 7.74239484902805277202583680305, 8.246516957838494741717718139746, 9.031967340409683125688161985301, 10.25513839899096579415801627891

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