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Properties

Label 2-770-385.142-c1-0-18
Degree $2$
Conductor $770$
Sign $0.916 - 0.400i$
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  + (0.258 + 0.965i)2-s + (−1.66 − 0.446i)3-s + (−0.866 + 0.499i)4-s + (−2.10 − 0.745i)5-s − 1.72i·6-s + (1.49 + 2.18i)7-s + (−0.707 − 0.707i)8-s + (−0.0154 − 0.00894i)9-s + (0.174 − 2.22i)10-s + (−1.04 − 3.14i)11-s + (1.66 − 0.446i)12-s + (2.15 + 2.15i)13-s + (−1.72 + 2.01i)14-s + (3.18 + 2.18i)15-s + (0.500 − 0.866i)16-s + (−2.94 − 0.788i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.963 − 0.258i)3-s + (−0.433 + 0.249i)4-s + (−0.942 − 0.333i)5-s − 0.704i·6-s + (0.565 + 0.824i)7-s + (−0.249 − 0.249i)8-s + (−0.00516 − 0.00298i)9-s + (0.0550 − 0.704i)10-s + (−0.313 − 0.949i)11-s + (0.481 − 0.129i)12-s + (0.598 + 0.598i)13-s + (−0.459 + 0.537i)14-s + (0.821 + 0.564i)15-s + (0.125 − 0.216i)16-s + (−0.713 − 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869816 + 0.181862i\)
\(L(\frac12)\) \(\approx\) \(0.869816 + 0.181862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (2.10 + 0.745i)T \)
7 \( 1 + (-1.49 - 2.18i)T \)
11 \( 1 + (1.04 + 3.14i)T \)
good3 \( 1 + (1.66 + 0.446i)T + (2.59 + 1.5i)T^{2} \)
13 \( 1 + (-2.15 - 2.15i)T + 13iT^{2} \)
17 \( 1 + (2.94 + 0.788i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.776 + 1.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.955 + 3.56i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.75T + 29T^{2} \)
31 \( 1 + (-3.11 - 5.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.25 + 1.94i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.18iT - 41T^{2} \)
43 \( 1 + (-6.93 - 6.93i)T + 43iT^{2} \)
47 \( 1 + (-1.73 + 0.464i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.78 - 2.35i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.01 - 2.31i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.39 + 4.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.01 + 15.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.05T + 71T^{2} \)
73 \( 1 + (-2.81 + 10.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.693 - 1.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.01 - 6.01i)T + 83iT^{2} \)
89 \( 1 + (12.4 + 7.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.27 + 6.27i)T + 97iT^{2} \)
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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

−10.82560268474970538270008577404, −9.047675867102835463246094755506, −8.604762033554357735423064990508, −7.82294078961771439898514567267, −6.63334542066038537842741681144, −6.05581889273274473972074206765, −5.06847994760137512424597856496, −4.37065764895243585231859776919, −2.91589148675101103978017847489, −0.74308236555054717545157844590, 0.859702716165102314564771299975, 2.66305232516404304173771918056, 4.08433546094548590515435993573, 4.54549049045634288344486834558, 5.61728065980639775355786970580, 6.71877160679162481007962248654, 7.76596669403391726493285943967, 8.424977890239295935203660602299, 9.926283320041059916883711102592, 10.48779849014418824145354852035

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