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Properties

Label 2-77-77.54-c3-0-15
Degree $2$
Conductor $77$
Sign $0.216 + 0.976i$
Analytic cond. $4.54314$
Root an. cond. $2.13146$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.44 − 0.833i)2-s + (6.44 − 3.71i)3-s + (−2.60 − 4.51i)4-s + (11.5 + 6.64i)5-s − 12.4·6-s + (12.2 − 13.9i)7-s + 22.0i·8-s + (14.1 − 24.5i)9-s + (−11.0 − 19.2i)10-s + (−36.4 + 0.686i)11-s + (−33.6 − 19.4i)12-s + 19.0·13-s + (−29.2 + 9.89i)14-s + 98.8·15-s + (−2.48 + 4.30i)16-s + (−25.2 − 43.7i)17-s + ⋯
L(s)  = 1  + (−0.510 − 0.294i)2-s + (1.23 − 0.715i)3-s + (−0.326 − 0.564i)4-s + (1.02 + 0.594i)5-s − 0.844·6-s + (0.660 − 0.751i)7-s + 0.974i·8-s + (0.524 − 0.908i)9-s + (−0.350 − 0.607i)10-s + (−0.999 + 0.0188i)11-s + (−0.808 − 0.466i)12-s + 0.407·13-s + (−0.558 + 0.188i)14-s + 1.70·15-s + (−0.0388 + 0.0672i)16-s + (−0.360 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(4.54314\)
Root analytic conductor: \(2.13146\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :3/2),\ 0.216 + 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.39719 - 1.12149i\)
\(L(\frac12)\) \(\approx\) \(1.39719 - 1.12149i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-12.2 + 13.9i)T \)
11 \( 1 + (36.4 - 0.686i)T \)
good2 \( 1 + (1.44 + 0.833i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-6.44 + 3.71i)T + (13.5 - 23.3i)T^{2} \)
5 \( 1 + (-11.5 - 6.64i)T + (62.5 + 108. i)T^{2} \)
13 \( 1 - 19.0T + 2.19e3T^{2} \)
17 \( 1 + (25.2 + 43.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-49.3 + 85.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (55.1 - 95.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 278. iT - 2.43e4T^{2} \)
31 \( 1 + (-54.5 + 31.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (200. - 347. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 127.T + 6.89e4T^{2} \)
43 \( 1 - 82.0iT - 7.95e4T^{2} \)
47 \( 1 + (-259. - 149. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (20.9 + 36.2i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-329. + 190. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (64.5 - 111. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (362. + 628. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 697.T + 3.57e5T^{2} \)
73 \( 1 + (-438. - 759. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (488. + 282. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 981.T + 5.71e5T^{2} \)
89 \( 1 + (460. + 265. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 774. iT - 9.12e5T^{2} \)
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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

−13.83749636732298498658524680325, −13.25825845332381404186555843677, −11.17621700330044629689179405288, −10.23668736155857957949671009508, −9.222561310845948972231082486136, −8.131300503409739878559655065334, −6.98021274519604736161297876655, −5.19916444874903978214523696499, −2.74334008157682190075715760169, −1.47678155440028877537072079666, 2.33587752877211489764265878667, 4.09604291496583643711833244545, 5.67500173708011518609098893926, 7.948773652955017842528709133332, 8.556832258045483843878594098912, 9.400540373093939736168650624365, 10.31434723089399417281099557039, 12.31057398382934268623952798805, 13.38255064958480126919616301451, 14.17140948951000173366476730534

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