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Properties

Label 2-770-385.139-c1-0-3
Degree $2$
Conductor $770$
Sign $-0.997 + 0.0665i$
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 + (−0.208 + 0.641i)3-s + (0.309 + 0.951i)4-s + (−1.77 + 1.36i)5-s + (−0.545 + 0.396i)6-s + (−1.13 − 2.38i)7-s + (−0.309 + 0.951i)8-s + (2.05 + 1.49i)9-s + (−2.23 + 0.0575i)10-s + (−2.00 + 2.64i)11-s − 0.674·12-s + (−1.40 + 1.92i)13-s + (0.482 − 2.60i)14-s + (−0.502 − 1.42i)15-s + (−0.809 + 0.587i)16-s + (−3.28 − 4.52i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.120 + 0.370i)3-s + (0.154 + 0.475i)4-s + (−0.793 + 0.608i)5-s + (−0.222 + 0.161i)6-s + (−0.430 − 0.902i)7-s + (−0.109 + 0.336i)8-s + (0.686 + 0.498i)9-s + (−0.706 + 0.0182i)10-s + (−0.604 + 0.796i)11-s − 0.194·12-s + (−0.388 + 0.534i)13-s + (0.128 − 0.695i)14-s + (−0.129 − 0.366i)15-s + (−0.202 + 0.146i)16-s + (−0.797 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.997 + 0.0665i$
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: \((2,\ 770,\ (\ :1/2),\ -0.997 + 0.0665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0303252 - 0.910011i\)
\(L(\frac12)\) \(\approx\) \(0.0303252 - 0.910011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 - 1.36i)T \)
7 \( 1 + (1.13 + 2.38i)T \)
11 \( 1 + (2.00 - 2.64i)T \)
good3 \( 1 + (0.208 - 0.641i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (1.40 - 1.92i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.28 + 4.52i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.51 - 4.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.09iT - 23T^{2} \)
29 \( 1 + (5.39 - 1.75i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.98 - 6.86i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.465 - 0.151i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.81 + 8.66i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.13T + 43T^{2} \)
47 \( 1 + (1.66 - 5.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.44 - 4.73i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-11.8 + 3.84i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.97 + 1.43i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + (-10.9 + 7.94i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.782 - 0.254i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.46 - 13.0i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.44 + 3.36i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + (-5.14 - 3.73i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.68328765492746864266397519981, −10.16102811584277363550479246710, −9.077616341940098595718391335282, −7.70056543619451745571896633998, −7.26172683223052660183652193464, −6.65505861826591997838779348979, −5.17210638059894716445646961989, −4.34257248226612737649781622991, −3.70763063369157401961372942543, −2.31427736273337005968737036841, 0.36532709761977901779137356399, 2.08478998360268392202553268047, 3.36036165873605899425756877418, 4.28157674169938719478250908600, 5.40530491727097767345376422436, 6.14615361517417011030024889709, 7.23461247208766849268017446456, 8.196733801626132731653260586763, 9.071130330315300720793110344170, 9.868950282078510034448616725629

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