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Properties

Label 2-82-41.32-c5-0-9
Degree $2$
Conductor $82$
Sign $0.837 - 0.546i$
Analytic cond. $13.1514$
Root an. cond. $3.62649$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (−2.52 + 2.52i)3-s − 16·4-s − 50.8i·5-s + (−10.1 − 10.1i)6-s + (−34.3 + 34.3i)7-s − 64i·8-s + 230. i·9-s + 203.·10-s + (439. − 439. i)11-s + (40.4 − 40.4i)12-s + (397. − 397. i)13-s + (−137. − 137. i)14-s + (128. + 128. i)15-s + 256·16-s + (516. + 516. i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.162 + 0.162i)3-s − 0.5·4-s − 0.909i·5-s + (−0.114 − 0.114i)6-s + (−0.265 + 0.265i)7-s − 0.353i·8-s + 0.947i·9-s + 0.642·10-s + (1.09 − 1.09i)11-s + (0.0811 − 0.0811i)12-s + (0.652 − 0.652i)13-s + (−0.187 − 0.187i)14-s + (0.147 + 0.147i)15-s + 0.250·16-s + (0.433 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(13.1514\)
Root analytic conductor: \(3.62649\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :5/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.61721 + 0.480563i\)
\(L(\frac12)\) \(\approx\) \(1.61721 + 0.480563i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
41 \( 1 + (-6.56e3 + 8.53e3i)T \)
good3 \( 1 + (2.52 - 2.52i)T - 243iT^{2} \)
5 \( 1 + 50.8iT - 3.12e3T^{2} \)
7 \( 1 + (34.3 - 34.3i)T - 1.68e4iT^{2} \)
11 \( 1 + (-439. + 439. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-397. + 397. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-516. - 516. i)T + 1.41e6iT^{2} \)
19 \( 1 + (-1.66e3 - 1.66e3i)T + 2.47e6iT^{2} \)
23 \( 1 + 436.T + 6.43e6T^{2} \)
29 \( 1 + (736. - 736. i)T - 2.05e7iT^{2} \)
31 \( 1 - 2.85e3T + 2.86e7T^{2} \)
37 \( 1 - 1.10e4T + 6.93e7T^{2} \)
43 \( 1 + 1.86e3iT - 1.47e8T^{2} \)
47 \( 1 + (1.95e4 + 1.95e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (-9.02e3 + 9.02e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 1.61e4T + 7.14e8T^{2} \)
61 \( 1 + 5.29e4iT - 8.44e8T^{2} \)
67 \( 1 + (-2.19e3 - 2.19e3i)T + 1.35e9iT^{2} \)
71 \( 1 + (3.52e3 - 3.52e3i)T - 1.80e9iT^{2} \)
73 \( 1 - 1.72e4iT - 2.07e9T^{2} \)
79 \( 1 + (5.98e4 - 5.98e4i)T - 3.07e9iT^{2} \)
83 \( 1 - 6.95e4T + 3.93e9T^{2} \)
89 \( 1 + (6.31e3 - 6.31e3i)T - 5.58e9iT^{2} \)
97 \( 1 + (7.88e4 + 7.88e4i)T + 8.58e9iT^{2} \)
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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

−13.56452341461659387946593822530, −12.56442572711899407948857609142, −11.32571737249229976466338490094, −9.918246363025102303563622341227, −8.701777175291256844755782809371, −7.924029716058994190845812909627, −6.12645313427060981846783417505, −5.26595462681338631320492048860, −3.69134978112481793837310599467, −1.02564770896850656547318170603, 1.12056018004477768940534980512, 3.00468843540949606176992506354, 4.29270048038915414879095330184, 6.34487129095638955399998698260, 7.23891012678136894417588845845, 9.204105014499746678909423841620, 9.878520622476071437022450856136, 11.33669854506555832889841945724, 11.88602370805469469072831379049, 13.14731907752488298240130238457

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