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Properties

Label 2-82-41.14-c4-0-8
Degree $2$
Conductor $82$
Sign $0.994 + 0.102i$
Analytic cond. $8.47633$
Root an. cond. $2.91141$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + (5.22 + 12.6i)3-s + 8i·4-s + (25.8 − 25.8i)5-s + (14.7 − 35.7i)6-s + (−20.1 − 48.5i)7-s + (16 − 16i)8-s + (−74.7 + 74.7i)9-s − 103.·10-s + (200. − 83.0i)11-s + (−100. + 41.8i)12-s + (84.8 + 204. i)13-s + (−56.9 + 137. i)14-s + (462. + 191. i)15-s − 64·16-s + (46.8 − 113. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.580 + 1.40i)3-s + 0.5i·4-s + (1.03 − 1.03i)5-s + (0.410 − 0.991i)6-s + (−0.410 − 0.991i)7-s + (0.250 − 0.250i)8-s + (−0.922 + 0.922i)9-s − 1.03·10-s + (1.65 − 0.686i)11-s + (−0.701 + 0.290i)12-s + (0.502 + 1.21i)13-s + (−0.290 + 0.701i)14-s + (2.05 + 0.850i)15-s − 0.250·16-s + (0.162 − 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.994 + 0.102i$
Analytic conductor: \(8.47633\)
Root analytic conductor: \(2.91141\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :2),\ 0.994 + 0.102i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.89129 - 0.0972160i\)
\(L(\frac12)\) \(\approx\) \(1.89129 - 0.0972160i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
41 \( 1 + (-1.38e3 + 959. i)T \)
good3 \( 1 + (-5.22 - 12.6i)T + (-57.2 + 57.2i)T^{2} \)
5 \( 1 + (-25.8 + 25.8i)T - 625iT^{2} \)
7 \( 1 + (20.1 + 48.5i)T + (-1.69e3 + 1.69e3i)T^{2} \)
11 \( 1 + (-200. + 83.0i)T + (1.03e4 - 1.03e4i)T^{2} \)
13 \( 1 + (-84.8 - 204. i)T + (-2.01e4 + 2.01e4i)T^{2} \)
17 \( 1 + (-46.8 + 113. i)T + (-5.90e4 - 5.90e4i)T^{2} \)
19 \( 1 + (116. - 280. i)T + (-9.21e4 - 9.21e4i)T^{2} \)
23 \( 1 + 490. iT - 2.79e5T^{2} \)
29 \( 1 + (-295. - 713. i)T + (-5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 - 599. iT - 9.23e5T^{2} \)
37 \( 1 - 706.T + 1.87e6T^{2} \)
43 \( 1 + (2.22e3 + 2.22e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (904. - 2.18e3i)T + (-3.45e6 - 3.45e6i)T^{2} \)
53 \( 1 + (1.95e3 - 809. i)T + (5.57e6 - 5.57e6i)T^{2} \)
59 \( 1 + 4.90e3T + 1.21e7T^{2} \)
61 \( 1 + (3.40e3 + 3.40e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-1.28e3 + 3.09e3i)T + (-1.42e7 - 1.42e7i)T^{2} \)
71 \( 1 + (-934. - 2.25e3i)T + (-1.79e7 + 1.79e7i)T^{2} \)
73 \( 1 + (-2.59e3 - 2.59e3i)T + 2.83e7iT^{2} \)
79 \( 1 + (7.62e3 - 3.15e3i)T + (2.75e7 - 2.75e7i)T^{2} \)
83 \( 1 + 1.43e3T + 4.74e7T^{2} \)
89 \( 1 + (3.21e3 + 7.76e3i)T + (-4.43e7 + 4.43e7i)T^{2} \)
97 \( 1 + (-1.46e4 - 6.08e3i)T + (6.25e7 + 6.25e7i)T^{2} \)
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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

−13.92788905302575420995046196815, −12.43390356272616991101784759031, −10.99905028528743662759539920191, −9.975364337680203155835041907796, −9.186021021114864321910197545859, −8.693243087466458792854078106927, −6.45620291490337731195428894957, −4.53367669092812452009732198224, −3.60066821207481027139877369722, −1.34471702366297509351637196239, 1.54615619006934141206643757706, 2.79621241642227259724391834719, 6.07935798625104053565575921984, 6.50210462946116033708534297929, 7.73419776217586115151687433463, 9.006617594063635671140388719987, 9.912446535838778309521186366892, 11.52864535128589732235973011804, 12.81510579731303989315861287135, 13.66504862562778800963038153012

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