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Properties

Label 2-770-385.139-c1-0-7
Degree $2$
Conductor $770$
Sign $-0.473 - 0.880i$
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 + (−0.266 + 0.820i)3-s + (0.309 + 0.951i)4-s + (1.93 + 1.11i)5-s + (0.697 − 0.507i)6-s + (−2.50 − 0.853i)7-s + (0.309 − 0.951i)8-s + (1.82 + 1.32i)9-s + (−0.908 − 2.04i)10-s + (−3.27 + 0.526i)11-s − 0.862·12-s + (−1.64 + 2.26i)13-s + (1.52 + 2.16i)14-s + (−1.43 + 1.28i)15-s + (−0.809 + 0.587i)16-s + (1.88 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.153 + 0.473i)3-s + (0.154 + 0.475i)4-s + (0.865 + 0.500i)5-s + (0.284 − 0.206i)6-s + (−0.946 − 0.322i)7-s + (0.109 − 0.336i)8-s + (0.608 + 0.442i)9-s + (−0.287 − 0.646i)10-s + (−0.987 + 0.158i)11-s − 0.249·12-s + (−0.455 + 0.627i)13-s + (0.407 + 0.577i)14-s + (−0.370 + 0.333i)15-s + (−0.202 + 0.146i)16-s + (0.456 + 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.473 - 0.880i$
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: \((2,\ 770,\ (\ :1/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383458 + 0.641813i\)
\(L(\frac12)\) \(\approx\) \(0.383458 + 0.641813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (2.50 + 0.853i)T \)
11 \( 1 + (3.27 - 0.526i)T \)
good3 \( 1 + (0.266 - 0.820i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (1.64 - 2.26i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.88 - 2.59i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.715 - 2.20i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.07iT - 23T^{2} \)
29 \( 1 + (4.94 - 1.60i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.37 + 3.27i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.11 - 1.33i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.541 - 1.66i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.38T + 43T^{2} \)
47 \( 1 + (2.33 - 7.17i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.07 - 1.48i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-9.20 + 2.98i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.40 - 4.64i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.62iT - 67T^{2} \)
71 \( 1 + (6.31 - 4.58i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.53 + 1.79i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.48 + 10.2i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.00 - 6.88i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + (2.05 + 1.49i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.34288384805069904549776530141, −9.972430321207607764041371345387, −9.299419690603646814460383874336, −8.077385030833681541645262365782, −7.14488453596193965737790759710, −6.35485587902783851080550348511, −5.23274888168989778861934057409, −4.05431124707617168399604185333, −2.89716184107922755807893821333, −1.81908316974579378328670092831, 0.44057965993894272043735998049, 1.96258220379480184570259084361, 3.23464987142471898161869606605, 5.08159302617028912137251620420, 5.67037919033428041063668420320, 6.64259409254071747460383975693, 7.34971346603830002958336719331, 8.335448263135996985181042639472, 9.355879002490204059094401609342, 9.826143305513909419912371370727

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