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Properties

Label 2-770-385.142-c1-0-19
Degree $2$
Conductor $770$
Sign $-0.781 - 0.624i$
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  + (0.258 + 0.965i)2-s + (2.51 + 0.674i)3-s + (−0.866 + 0.499i)4-s + (0.550 + 2.16i)5-s + 2.60i·6-s + (−2.23 + 1.41i)7-s + (−0.707 − 0.707i)8-s + (3.28 + 1.89i)9-s + (−1.95 + 1.09i)10-s + (−3.21 + 0.831i)11-s + (−2.51 + 0.674i)12-s + (0.233 + 0.233i)13-s + (−1.94 − 1.79i)14-s + (−0.0752 + 5.82i)15-s + (0.500 − 0.866i)16-s + (1.03 + 0.278i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (1.45 + 0.389i)3-s + (−0.433 + 0.249i)4-s + (0.246 + 0.969i)5-s + 1.06i·6-s + (−0.845 + 0.533i)7-s + (−0.249 − 0.249i)8-s + (1.09 + 0.632i)9-s + (−0.616 + 0.345i)10-s + (−0.968 + 0.250i)11-s + (−0.726 + 0.194i)12-s + (0.0647 + 0.0647i)13-s + (−0.519 − 0.479i)14-s + (−0.0194 + 1.50i)15-s + (0.125 − 0.216i)16-s + (0.251 + 0.0674i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.781 - 0.624i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.756197 + 2.15827i\)
\(L(\frac12)\) \(\approx\) \(0.756197 + 2.15827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-0.550 - 2.16i)T \)
7 \( 1 + (2.23 - 1.41i)T \)
11 \( 1 + (3.21 - 0.831i)T \)
good3 \( 1 + (-2.51 - 0.674i)T + (2.59 + 1.5i)T^{2} \)
13 \( 1 + (-0.233 - 0.233i)T + 13iT^{2} \)
17 \( 1 + (-1.03 - 0.278i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.68 + 2.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.270 - 1.00i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 + (-1.19 - 2.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-11.1 + 2.99i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.15iT - 41T^{2} \)
43 \( 1 + (0.522 + 0.522i)T + 43iT^{2} \)
47 \( 1 + (5.98 - 1.60i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.89 - 1.58i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.71 + 0.987i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.62 - 4.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.30 - 12.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (1.14 - 4.26i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.50 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.47 - 8.47i)T + 83iT^{2} \)
89 \( 1 + (13.1 + 7.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.281 + 0.281i)T + 97iT^{2} \)
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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

−10.12703286698618044490849498911, −9.769056305753247583634680886370, −8.887435600289207969568006812508, −8.069734393593533516556746582630, −7.28557518714037931780457484479, −6.42282307219554887402533159176, −5.38667946005260317583439319624, −4.09378828972829334908896145663, −2.96747298193250237450417126593, −2.59171484914933563074120388875, 0.942299373027332050214963036040, 2.34347944713175547277673376745, 3.20324404320009617749259405596, 4.15638837185025554958069260931, 5.31840947524693662170125618694, 6.50773418632911229091496362175, 7.906928225624452302304226981082, 8.197346999593638449790249226102, 9.301116372475978553126529085891, 9.780094257044517532169113418622

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