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Properties

Label 2-77-11.3-c1-0-3
Degree $2$
Conductor $77$
Sign $0.986 + 0.160i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
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  + (1.38 − 1.00i)2-s + (0.708 + 2.17i)3-s + (0.282 − 0.869i)4-s + (−3.28 − 2.39i)5-s + (3.16 + 2.29i)6-s + (0.309 − 0.951i)7-s + (0.572 + 1.76i)8-s + (−1.82 + 1.32i)9-s − 6.94·10-s + (−0.582 − 3.26i)11-s + 2.09·12-s + (−2.65 + 1.92i)13-s + (−0.527 − 1.62i)14-s + (2.87 − 8.86i)15-s + (4.03 + 2.93i)16-s + (1.06 + 0.776i)17-s + ⋯
L(s)  = 1  + (0.976 − 0.709i)2-s + (0.408 + 1.25i)3-s + (0.141 − 0.434i)4-s + (−1.47 − 1.06i)5-s + (1.29 + 0.938i)6-s + (0.116 − 0.359i)7-s + (0.202 + 0.623i)8-s + (−0.607 + 0.441i)9-s − 2.19·10-s + (−0.175 − 0.984i)11-s + 0.604·12-s + (−0.735 + 0.534i)13-s + (−0.140 − 0.433i)14-s + (0.743 − 2.28i)15-s + (1.00 + 0.733i)16-s + (0.259 + 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.986 + 0.160i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.986 + 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34755 - 0.109068i\)
\(L(\frac12)\) \(\approx\) \(1.34755 - 0.109068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.582 + 3.26i)T \)
good2 \( 1 + (-1.38 + 1.00i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.708 - 2.17i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (3.28 + 2.39i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.65 - 1.92i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.06 - 0.776i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.668 - 2.05i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + (0.0754 - 0.232i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.55 + 4.03i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.0789 + 0.243i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.77 + 5.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (1.25 + 3.86i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.04 + 2.94i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.303 - 0.935i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.49 + 1.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + (5.23 + 3.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.98 - 9.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.47 + 3.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.67 + 1.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (-2.09 + 1.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.45173229385698685226365872166, −13.33388999134886432437339770922, −12.11640979977259148804391550176, −11.49200664702851767149523479351, −10.27418576167910057529001011927, −8.781357073696103130037554380339, −7.905331872912529186491208630678, −5.11331256105866516731032227682, −4.21801943229774669434406899361, −3.45841311397960636141667034005, 2.96459395047780766182849500593, 4.68509719340527348716189789709, 6.56975798705602970615245472346, 7.33102457271540089722332660942, 7.996064191372964738282258281173, 10.20348461614814954853535023057, 11.86925815352722376638724362520, 12.46140136727319049361961122833, 13.59124545903935397432213528349, 14.67616623978294296193133249536

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