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Properties

Label 2-770-385.164-c1-0-44
Degree $2$
Conductor $770$
Sign $-0.992 + 0.118i$
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 + (1.64 − 2.85i)3-s + (−0.499 + 0.866i)4-s + (1.92 + 1.13i)5-s − 3.29·6-s + (−1.01 − 2.44i)7-s + 0.999·8-s + (−3.93 − 6.81i)9-s + (0.0163 − 2.23i)10-s + (−3.06 − 1.27i)11-s + (1.64 + 2.85i)12-s + 3.27i·13-s + (−1.60 + 2.10i)14-s + (6.40 − 3.63i)15-s + (−0.5 − 0.866i)16-s + (1.23 + 0.711i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.951 − 1.64i)3-s + (−0.249 + 0.433i)4-s + (0.862 + 0.506i)5-s − 1.34·6-s + (−0.383 − 0.923i)7-s + 0.353·8-s + (−1.31 − 2.27i)9-s + (0.00517 − 0.707i)10-s + (−0.923 − 0.383i)11-s + (0.475 + 0.824i)12-s + 0.908i·13-s + (−0.430 + 0.561i)14-s + (1.65 − 0.939i)15-s + (−0.125 − 0.216i)16-s + (0.298 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.992 + 0.118i$
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: \((2,\ 770,\ (\ :1/2),\ -0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0965109 - 1.62369i\)
\(L(\frac12)\) \(\approx\) \(0.0965109 - 1.62369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.92 - 1.13i)T \)
7 \( 1 + (1.01 + 2.44i)T \)
11 \( 1 + (3.06 + 1.27i)T \)
good3 \( 1 + (-1.64 + 2.85i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 3.27iT - 13T^{2} \)
17 \( 1 + (-1.23 - 0.711i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 + 3.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.95 + 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.06iT - 29T^{2} \)
31 \( 1 + (4.72 + 2.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.46 + 3.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 - 3.88T + 43T^{2} \)
47 \( 1 + (6.42 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.28 - 3.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.91 - 5.72i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.842 + 1.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.99 - 2.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.01T + 71T^{2} \)
73 \( 1 + (-11.7 - 6.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.362 + 0.209i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + (7.47 - 4.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.51T + 97T^{2} \)
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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

−9.747217994267372845013140115622, −9.076902126735141338770853089514, −8.177228636685553628049725170935, −7.27838413437362439581654940755, −6.80924822824439477362203560087, −5.78582716699393358850474167697, −3.89468207162091003027667286779, −2.75029334909096030099291301726, −2.13816930366307320870836504230, −0.808526305030981803252145883896, 2.28021804232641750855897880407, 3.22518518307760355226392104629, 4.64648246000113222385812793067, 5.35272779499726484523701890285, 5.95903007034291156252209140784, 7.71799799341188960065471340223, 8.403326009420408264420295298268, 9.133369769921135426085315847467, 9.712244452666698128066437525277, 10.24888161775987145680944757888

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