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Properties

Label 2-770-385.164-c1-0-1
Degree $2$
Conductor $770$
Sign $-0.998 + 0.0590i$
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 + (−0.438 + 2.19i)5-s + 1.56·6-s + (−0.725 − 2.54i)7-s + 0.999·8-s + (0.278 + 0.481i)9-s + (2.11 − 0.716i)10-s + (−2.06 + 2.59i)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 + (−0.196 + 0.980i)5-s + 0.638·6-s + (−0.274 − 0.961i)7-s + 0.353·8-s + (0.0927 + 0.160i)9-s + (0.669 − 0.226i)10-s + (−0.622 + 0.782i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.998 + 0.0590i$
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: \((2,\ 770,\ (\ :1/2),\ -0.998 + 0.0590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00656150 - 0.222026i\)
\(L(\frac12)\) \(\approx\) \(0.00656150 - 0.222026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.438 - 2.19i)T \)
7 \( 1 + (0.725 + 2.54i)T \)
11 \( 1 + (2.06 - 2.59i)T \)
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

−10.70730523863254618343103547270, −10.03033136005101609980557221418, −9.559410340476163974040719948909, −8.226898507199321537122341970214, −7.16396729630263958815534333796, −6.72577910756032664545145454828, −5.02883780669202748945067982021, −4.32327280306086496999402787477, −3.33993956012124565952908271063, −2.09108311219360785107903015497, 0.13393315274629904893453021573, 1.50527650166177225011034394569, 3.19564798687817653668237315009, 4.90977161366641240425194285592, 5.50703048012736413052224939709, 6.41195587314604447300430966701, 7.22094265261214939715612864713, 8.262256586025582965119240348742, 8.881079527116046350729064793596, 9.488559137063960159253524871027

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