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Properties

Label 2-770-385.349-c1-0-12
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: $\chi_{770} (349, \cdot )$
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

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

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