Properties

Label 2-770-385.164-c1-0-46
Degree $2$
Conductor $770$
Sign $-0.224 - 0.974i$
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 + (0.781 − 1.35i)3-s + (−0.499 + 0.866i)4-s + (−2.11 − 0.716i)5-s − 1.56·6-s + (−0.725 − 2.54i)7-s + 0.999·8-s + (0.278 + 0.481i)9-s + (0.438 + 2.19i)10-s + (−1.21 + 3.08i)11-s + (0.781 + 1.35i)12-s + 2.41i·13-s + (−1.84 + 1.90i)14-s + (−2.62 + 2.30i)15-s + (−0.5 − 0.866i)16-s + (−6.35 − 3.66i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.451 − 0.781i)3-s + (−0.249 + 0.433i)4-s + (−0.947 − 0.320i)5-s − 0.638·6-s + (−0.274 − 0.961i)7-s + 0.353·8-s + (0.0927 + 0.160i)9-s + (0.138 + 0.693i)10-s + (−0.366 + 0.930i)11-s + (0.225 + 0.390i)12-s + 0.670i·13-s + (−0.491 + 0.507i)14-s + (−0.677 + 0.595i)15-s + (−0.125 − 0.216i)16-s + (−1.54 − 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.224 - 0.974i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0681013 + 0.0855544i\)
\(L(\frac12)\) \(\approx\) \(0.0681013 + 0.0855544i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.11 + 0.716i)T \)
7 \( 1 + (0.725 + 2.54i)T \)
11 \( 1 + (1.21 - 3.08i)T \)
good3 \( 1 + (-0.781 + 1.35i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 2.41iT - 13T^{2} \)
17 \( 1 + (6.35 + 3.66i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.89 + 5.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.48 - 4.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 + (0.862 + 0.497i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.20 + 1.85i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.97T + 41T^{2} \)
43 \( 1 + 6.18T + 43T^{2} \)
47 \( 1 + (-5.29 - 9.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.54 - 0.889i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.38 + 4.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.71 + 2.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.76 + 5.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.788T + 71T^{2} \)
73 \( 1 + (8.04 + 4.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.68 - 1.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (3.28 - 1.89i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.37T + 97T^{2} \)
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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

−9.589544900685405634770965270741, −8.906065283671353464258532259431, −7.85563639709298894331055261684, −7.33567838991149388463080876499, −6.71207511569296513150363112608, −4.62219509873744976389899050279, −4.23985492211982561556041394836, −2.76704865016756546941372433623, −1.67799324823671250704321318585, −0.05665691733118686825790204794, 2.50502617545140630158408365232, 3.73443016357111864812098159664, 4.42926994967464512356780328566, 5.91529289856650359834460373422, 6.39461295263251484471487254325, 7.80791363893784100500920905201, 8.522801030723648088528601112081, 8.842456182213504877782737385858, 10.17952219048830174582507086456, 10.51599909117922759088221873451

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