Properties

Label 2-770-11.4-c1-0-9
Degree $2$
Conductor $770$
Sign $0.985 - 0.172i$
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 + (−0.0652 + 0.200i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.170 − 0.124i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (2.39 + 1.73i)9-s − 10-s + (−0.544 + 3.27i)11-s − 0.211·12-s + (0.310 + 0.225i)13-s + (−0.309 + 0.951i)14-s + (0.0652 + 0.200i)15-s + (−0.809 + 0.587i)16-s + (2.71 − 1.96i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.0376 + 0.115i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.0697 − 0.0506i)6-s + (−0.116 − 0.359i)7-s + (0.109 − 0.336i)8-s + (0.797 + 0.579i)9-s − 0.316·10-s + (−0.164 + 0.986i)11-s − 0.0609·12-s + (0.0861 + 0.0626i)13-s + (−0.0825 + 0.254i)14-s + (0.0168 + 0.0518i)15-s + (−0.202 + 0.146i)16-s + (0.657 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.985 - 0.172i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.985 - 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25056 + 0.108455i\)
\(L(\frac12)\) \(\approx\) \(1.25056 + 0.108455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.544 - 3.27i)T \)
good3 \( 1 + (0.0652 - 0.200i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-0.310 - 0.225i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.71 + 1.96i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.48 - 4.58i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.683T + 23T^{2} \)
29 \( 1 + (-0.496 - 1.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.01 - 0.739i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.264 - 0.814i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.97 + 9.15i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.55T + 43T^{2} \)
47 \( 1 + (2.33 - 7.18i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-10.9 - 7.95i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.24 - 6.91i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.14 - 2.28i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.97T + 67T^{2} \)
71 \( 1 + (0.679 - 0.493i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.19 + 9.84i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.72 + 1.97i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.51 + 6.18i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + (9.11 + 6.62i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.29655371300808024114451633132, −9.665872002957659568473169946037, −8.832411137560077625790283699384, −7.64002431835759318654110953245, −7.25743776360039553798072990193, −5.93862896044040217619477727081, −4.76995899675451538264687685252, −3.89961038935394825154033375027, −2.41642179482555713401762499222, −1.29778826921796046356221761234, 0.915944889046582870091855488884, 2.46847932342968312964028477367, 3.76274052224334342123607216633, 5.17388818695793690243730249608, 6.13517699068878285905771589696, 6.72799903024356144290280732330, 7.75343242678717967871963200705, 8.621162149677873854761198706263, 9.414987966515716894552061686911, 10.13750152293958833419970600261

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