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Properties

Label 2-770-385.139-c1-0-9
Degree $2$
Conductor $770$
Sign $-0.804 + 0.593i$
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 + (−2.63 − 0.282i)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 + (−1.96 − 1.77i)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 + (−0.994 − 0.106i)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 + (−0.524 − 0.474i)14-s + (−1.36 − 0.143i)15-s + (−0.202 + 0.146i)16-s + (0.0907 + 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.804 + 0.593i$
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: \((2,\ 770,\ (\ :1/2),\ -0.804 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360149 - 1.09475i\)
\(L(\frac12)\) \(\approx\) \(0.360149 - 1.09475i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.282i)T \)
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

−10.71290031880027020604748143025, −10.08572188837294924316130191320, −9.455199053305187590706899502934, −8.178797932921780884033889391671, −7.17036879149708496153777579957, −6.11356149061310553023131834200, −5.65981743163942836581044638875, −4.49169443886104983053848996524, −3.48651866039281379604027402060, −2.84266312567105145645141754457, 0.48842751203170285159207299259, 1.80970456226724300032485366283, 2.91171855453269030951509149798, 4.46903770081945696697549816568, 5.34090746612608415191780982066, 6.33577654132599769791891108376, 6.87798169684381126832333881017, 8.007105282413981831502634549330, 8.955245233017021289915135249404, 9.954418209783793409634114649202

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