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Properties

Label 2-77-77.62-c1-0-0
Degree $2$
Conductor $77$
Sign $-0.0349 - 0.999i$
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  + (−0.395 + 0.544i)2-s + (−2.52 − 0.819i)3-s + (0.478 + 1.47i)4-s + (2.08 + 2.87i)5-s + (1.44 − 1.04i)6-s + (−1.94 + 1.79i)7-s + (−2.26 − 0.737i)8-s + (3.26 + 2.37i)9-s − 2.38·10-s + (2.70 − 1.91i)11-s − 4.10i·12-s + (−0.892 − 0.648i)13-s + (−0.205 − 1.76i)14-s + (−2.90 − 8.95i)15-s + (−1.20 + 0.875i)16-s + (3.37 − 2.45i)17-s + ⋯
L(s)  = 1  + (−0.279 + 0.384i)2-s + (−1.45 − 0.473i)3-s + (0.239 + 0.735i)4-s + (0.933 + 1.28i)5-s + (0.589 − 0.428i)6-s + (−0.735 + 0.677i)7-s + (−0.802 − 0.260i)8-s + (1.08 + 0.790i)9-s − 0.755·10-s + (0.815 − 0.578i)11-s − 1.18i·12-s + (−0.247 − 0.179i)13-s + (−0.0550 − 0.472i)14-s + (−0.751 − 2.31i)15-s + (−0.301 + 0.218i)16-s + (0.818 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0349 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.0349 - 0.999i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.0349 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419430 + 0.434339i\)
\(L(\frac12)\) \(\approx\) \(0.419430 + 0.434339i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.94 - 1.79i)T \)
11 \( 1 + (-2.70 + 1.91i)T \)
good2 \( 1 + (0.395 - 0.544i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.52 + 0.819i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.08 - 2.87i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (0.892 + 0.648i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.37 + 2.45i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.384 - 1.18i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + (-1.94 + 0.631i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.76 + 2.42i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.853 - 2.62i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.69 + 5.22i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (-6.49 - 2.11i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.873 + 0.634i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.17 - 0.707i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.8 - 7.84i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.489T + 67T^{2} \)
71 \( 1 + (8.25 - 5.99i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.87 + 11.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.30 + 10.0i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 2.30iT - 89T^{2} \)
97 \( 1 + (7.44 - 10.2i)T + (-29.9 - 92.2i)T^{2} \)
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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

−14.89896893485376070029553643573, −13.54355673167466594033074897625, −12.31863585499465395260372546003, −11.64444919908241165722340127239, −10.48780831795410485031804940448, −9.237711700965540142142476040655, −7.32074665942557501955039397315, −6.39545391992943816399616245875, −5.83833674250129580209652355499, −2.96608923847676572040734488469, 1.13784757730233213702691431889, 4.61587555788785277451312145825, 5.69899494523923008519290241835, 6.62097405216340633434461286490, 9.205897116040165912431722020495, 9.896752261153113702675690653128, 10.69924524853161742362162015331, 11.98866321910926607379881965075, 12.74122382880143544346425740831, 14.14512233084188113120953927830

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