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Properties

Label 2-77-11.9-c1-0-4
Degree $2$
Conductor $77$
Sign $0.0851 + 0.996i$
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  + (0.549 − 1.69i)2-s + (−0.5 + 0.363i)3-s + (−0.938 − 0.681i)4-s + (−0.858 − 2.64i)5-s + (0.339 + 1.04i)6-s + (0.809 + 0.587i)7-s + (1.20 − 0.878i)8-s + (−0.809 + 2.48i)9-s − 4.93·10-s + (1.14 + 3.11i)11-s + 0.716·12-s + (−1.32 + 4.08i)13-s + (1.43 − 1.04i)14-s + (1.38 + 1.00i)15-s + (−1.53 − 4.73i)16-s + (−0.851 − 2.62i)17-s + ⋯
L(s)  = 1  + (0.388 − 1.19i)2-s + (−0.288 + 0.209i)3-s + (−0.469 − 0.340i)4-s + (−0.383 − 1.18i)5-s + (0.138 + 0.426i)6-s + (0.305 + 0.222i)7-s + (0.427 − 0.310i)8-s + (−0.269 + 0.829i)9-s − 1.56·10-s + (0.346 + 0.938i)11-s + 0.206·12-s + (−0.367 + 1.13i)13-s + (0.384 − 0.279i)14-s + (0.358 + 0.260i)15-s + (−0.384 − 1.18i)16-s + (−0.206 − 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0851 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0851 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.0851 + 0.996i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.0851 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770580 - 0.707569i\)
\(L(\frac12)\) \(\approx\) \(0.770580 - 0.707569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-1.14 - 3.11i)T \)
good2 \( 1 + (-0.549 + 1.69i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.5 - 0.363i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.858 + 2.64i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.32 - 4.08i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.851 + 2.62i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.56 - 1.13i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.37T + 23T^{2} \)
29 \( 1 + (6.98 + 5.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.0619 - 0.190i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.837 + 0.608i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.77 - 5.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (-10.5 + 7.66i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.20 - 3.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.92 - 5.02i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.305 + 0.940i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.41T + 67T^{2} \)
71 \( 1 + (0.623 + 1.91i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.06 - 5.86i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.94 + 5.98i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.531 + 1.63i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (-3.58 + 11.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.82209331733442013414830155731, −12.85721238210164139116535711129, −11.89049357861566402135258070758, −11.36625319345105271347646498217, −9.981151984861671410335448186762, −8.853576664427493305403330420548, −7.31401205631297440546608844201, −5.01508108947484100816107803017, −4.28011407875480074429386371100, −2.00084791067807242737199077908, 3.48409995540354735534011642463, 5.47522674241852729326163910318, 6.55853959900630711951262498520, 7.37179068209131051913782203376, 8.642206322804861865338274755464, 10.65348192571088045895263895535, 11.29717755213278028493837787177, 12.84164348666948693474619490737, 14.10290014596542861854925155937, 14.96386703265540001029261997768

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