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Properties

Label 2-770-385.142-c1-0-2
Degree $2$
Conductor $770$
Sign $-0.225 + 0.974i$
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 + (−1.43 − 0.383i)3-s + (−0.866 + 0.499i)4-s + (−0.782 + 2.09i)5-s − 1.48i·6-s + (−1.67 + 2.04i)7-s + (−0.707 − 0.707i)8-s + (−0.695 − 0.401i)9-s + (−2.22 − 0.213i)10-s + (2.76 + 1.83i)11-s + (1.43 − 0.383i)12-s + (1.48 + 1.48i)13-s + (−2.40 − 1.09i)14-s + (1.92 − 2.69i)15-s + (0.500 − 0.866i)16-s + (−4.96 − 1.33i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.826 − 0.221i)3-s + (−0.433 + 0.249i)4-s + (−0.349 + 0.936i)5-s − 0.605i·6-s + (−0.634 + 0.773i)7-s + (−0.249 − 0.249i)8-s + (−0.231 − 0.133i)9-s + (−0.703 − 0.0675i)10-s + (0.833 + 0.552i)11-s + (0.413 − 0.110i)12-s + (0.412 + 0.412i)13-s + (−0.644 − 0.291i)14-s + (0.496 − 0.696i)15-s + (0.125 − 0.216i)16-s + (−1.20 − 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.225 + 0.974i$
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: \((2,\ 770,\ (\ :1/2),\ -0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0615213 - 0.0774142i\)
\(L(\frac12)\) \(\approx\) \(0.0615213 - 0.0774142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.782 - 2.09i)T \)
7 \( 1 + (1.67 - 2.04i)T \)
11 \( 1 + (-2.76 - 1.83i)T \)
good3 \( 1 + (1.43 + 0.383i)T + (2.59 + 1.5i)T^{2} \)
13 \( 1 + (-1.48 - 1.48i)T + 13iT^{2} \)
17 \( 1 + (4.96 + 1.33i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.857 - 1.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.79 + 6.70i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.21T + 29T^{2} \)
31 \( 1 + (5.12 + 8.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.80 - 0.750i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.07iT - 41T^{2} \)
43 \( 1 + (2.15 + 2.15i)T + 43iT^{2} \)
47 \( 1 + (0.762 - 0.204i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (11.5 + 3.10i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.533 + 0.308i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.59 - 3.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.74 + 6.51i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.839T + 71T^{2} \)
73 \( 1 + (4.27 - 15.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.61 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.25 + 7.25i)T + 83iT^{2} \)
89 \( 1 + (7.52 + 4.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.25 - 8.25i)T + 97iT^{2} \)
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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

−11.13585300005811561522187364843, −10.05253013755181616263745871370, −9.107135427928083733544244712972, −8.336584164052862736313235606552, −7.04144044388413761897170736144, −6.38252375359152721542038579941, −6.14748961602796009658072829986, −4.74683799476984575346877713129, −3.72111822442139103350450472652, −2.41364746578605937105438476620, 0.05502327078407300373191784161, 1.35770857388526539413503251064, 3.30248476368552609258942254048, 4.15698421634546649288194085868, 5.05083850946698896525068579363, 5.97006316031184122994016026179, 6.88728932216838261032591205749, 8.291332006363927256341264354695, 8.953368228175015280433153941213, 9.860105231044718857128067270259

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