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Properties

Label 2-770-11.9-c1-0-10
Degree $2$
Conductor $770$
Sign $0.0219 - 0.999i$
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  + (−0.309 + 0.951i)2-s + (−2.11 + 1.53i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.809 − 2.48i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (1.19 − 3.66i)9-s + 0.999·10-s + (−0.809 − 3.21i)11-s + 2.61·12-s + (−0.618 + 1.90i)13-s + (−0.809 + 0.587i)14-s + (2.11 + 1.53i)15-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.22 + 0.888i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (−0.330 − 1.01i)6-s + (0.305 + 0.222i)7-s + (0.286 − 0.207i)8-s + (0.396 − 1.22i)9-s + 0.316·10-s + (−0.243 − 0.969i)11-s + 0.755·12-s + (−0.171 + 0.527i)13-s + (−0.216 + 0.157i)14-s + (0.546 + 0.397i)15-s + (0.0772 + 0.237i)16-s + (0.121 + 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576144 + 0.563614i\)
\(L(\frac12)\) \(\approx\) \(0.576144 + 0.563614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.809 + 3.21i)T \)
good3 \( 1 + (2.11 - 1.53i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (0.618 - 1.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.54 + 4.02i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-2.61 - 1.90i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.381 - 1.17i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.23 - 3.80i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.35 - 6.06i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (7.47 - 5.42i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.381 + 1.17i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.16 - 4.47i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.09 - 3.35i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + (3 + 9.23i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.2 - 8.14i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.61 + 8.05i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.19 + 9.82i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + (2.95 - 9.09i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.51419102103151988823765723764, −9.638352257338205082673283814764, −8.906329929619231597027010529019, −8.022469941703177938172310311690, −6.84834675328297815007934050472, −6.00610576834130099231341737649, −5.02868765053528121284772560271, −4.78372030970651450076704902417, −3.30949161081998482067016729011, −0.915799072505823807961652277666, 0.75985646841372947604008526393, 2.01741920896444142621171316237, 3.40902918521480707039789427746, 4.87812660532622234742415692350, 5.53459122344021600823175703343, 6.82250623746700134194754160944, 7.38394596277946732537898082954, 8.184894810175326001416995750522, 9.643946203946432941898016823641, 10.25939683155703377655648265700

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