Properties

Label 2-770-385.139-c1-0-44
Degree $2$
Conductor $770$
Sign $-0.0927 + 0.995i$
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  + (0.809 + 0.587i)2-s + (0.732 − 2.25i)3-s + (0.309 + 0.951i)4-s + (−0.464 − 2.18i)5-s + (1.91 − 1.39i)6-s + (1.96 − 1.77i)7-s + (−0.309 + 0.951i)8-s + (−2.12 − 1.54i)9-s + (0.910 − 2.04i)10-s + (−2.63 − 2.01i)11-s + 2.37·12-s + (−0.478 + 0.657i)13-s + (2.63 − 0.282i)14-s + (−5.27 − 0.556i)15-s + (−0.809 + 0.587i)16-s + (−0.374 − 0.514i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.423 − 1.30i)3-s + (0.154 + 0.475i)4-s + (−0.207 − 0.978i)5-s + (0.783 − 0.569i)6-s + (0.741 − 0.670i)7-s + (−0.109 + 0.336i)8-s + (−0.708 − 0.514i)9-s + (0.287 − 0.645i)10-s + (−0.793 − 0.608i)11-s + 0.684·12-s + (−0.132 + 0.182i)13-s + (0.703 − 0.0754i)14-s + (−1.36 − 0.143i)15-s + (−0.202 + 0.146i)16-s + (−0.0907 − 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0927 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0927 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.0927 + 0.995i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.0927 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55049 - 1.70165i\)
\(L(\frac12)\) \(\approx\) \(1.55049 - 1.70165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.464 + 2.18i)T \)
7 \( 1 + (-1.96 + 1.77i)T \)
11 \( 1 + (2.63 + 2.01i)T \)
good3 \( 1 + (-0.732 + 2.25i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (0.478 - 0.657i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.374 + 0.514i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.231 + 0.711i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.84iT - 23T^{2} \)
29 \( 1 + (-8.07 + 2.62i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.72 - 3.75i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.85 - 1.25i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.693 + 2.13i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + (-1.40 + 4.31i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.23 + 1.70i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-11.4 + 3.71i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-9.48 + 6.88i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.11iT - 67T^{2} \)
71 \( 1 + (-6.16 + 4.47i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.50 - 3.08i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.15 - 12.6i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.36 - 6.00i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 + (-9.78 - 7.10i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.08930180653881406648356967253, −8.664713822690972440680054549543, −8.233565848791593349099722327580, −7.48696180939631964291319849680, −6.83415652157060892919685736964, −5.55741838003602368687504598944, −4.82376786946079070762316186820, −3.63332467985701933427504529001, −2.21090693389294159697496436678, −0.958086244931515785891621424812, 2.31294136124997981261572410241, 3.02136673105132040147461031041, 4.18726857805395897654956736331, 4.84991553318670521167215058342, 5.82287378508922676640170197061, 7.04030745867516766218476194797, 8.135467786484531674376669555659, 8.987318123103367998454799507105, 10.12797234508414163765374345106, 10.40109692735043142550523683840

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