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Properties

Label 2-82-41.3-c4-0-10
Degree $2$
Conductor $82$
Sign $-0.888 + 0.458i$
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 + (5.35 − 12.9i)3-s − 8i·4-s + (−14.6 − 14.6i)5-s + (15.1 + 36.5i)6-s + (11.3 − 27.5i)7-s + (16 + 16i)8-s + (−81.2 − 81.2i)9-s + 58.5·10-s + (−40.0 − 16.5i)11-s + (−103. − 42.8i)12-s + (−116. + 280. i)13-s + (32.2 + 77.8i)14-s + (−267. + 110. i)15-s − 64·16-s + (48.6 + 117. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.595 − 1.43i)3-s − 0.5i·4-s + (−0.585 − 0.585i)5-s + (0.420 + 1.01i)6-s + (0.232 − 0.561i)7-s + (0.250 + 0.250i)8-s + (−1.00 − 1.00i)9-s + 0.585·10-s + (−0.330 − 0.136i)11-s + (−0.718 − 0.297i)12-s + (−0.686 + 1.65i)13-s + (0.164 + 0.397i)14-s + (−1.18 + 0.492i)15-s − 0.250·16-s + (0.168 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $-0.888 + 0.458i$
Analytic conductor: \(8.47633\)
Root analytic conductor: \(2.91141\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :2),\ -0.888 + 0.458i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.223085 - 0.918202i\)
\(L(\frac12)\) \(\approx\) \(0.223085 - 0.918202i\)
\(L(3)\) 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 \( 1 + (1.63e3 + 389. i)T \)
good3 \( 1 + (-5.35 + 12.9i)T + (-57.2 - 57.2i)T^{2} \)
5 \( 1 + (14.6 + 14.6i)T + 625iT^{2} \)
7 \( 1 + (-11.3 + 27.5i)T + (-1.69e3 - 1.69e3i)T^{2} \)
11 \( 1 + (40.0 + 16.5i)T + (1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + (116. - 280. i)T + (-2.01e4 - 2.01e4i)T^{2} \)
17 \( 1 + (-48.6 - 117. i)T + (-5.90e4 + 5.90e4i)T^{2} \)
19 \( 1 + (257. + 622. i)T + (-9.21e4 + 9.21e4i)T^{2} \)
23 \( 1 + 141. iT - 2.79e5T^{2} \)
29 \( 1 + (341. - 823. i)T + (-5.00e5 - 5.00e5i)T^{2} \)
31 \( 1 + 1.25e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.03e3T + 1.87e6T^{2} \)
43 \( 1 + (-626. + 626. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-184. - 445. i)T + (-3.45e6 + 3.45e6i)T^{2} \)
53 \( 1 + (1.31e3 + 544. i)T + (5.57e6 + 5.57e6i)T^{2} \)
59 \( 1 - 3.75e3T + 1.21e7T^{2} \)
61 \( 1 + (-1.25e3 + 1.25e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (2.34e3 + 5.65e3i)T + (-1.42e7 + 1.42e7i)T^{2} \)
71 \( 1 + (-1.61e3 + 3.91e3i)T + (-1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (-4.49e3 + 4.49e3i)T - 2.83e7iT^{2} \)
79 \( 1 + (1.14e3 + 475. i)T + (2.75e7 + 2.75e7i)T^{2} \)
83 \( 1 - 170.T + 4.74e7T^{2} \)
89 \( 1 + (4.13e3 - 9.98e3i)T + (-4.43e7 - 4.43e7i)T^{2} \)
97 \( 1 + (-7.73e3 + 3.20e3i)T + (6.25e7 - 6.25e7i)T^{2} \)
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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

−13.26015343950845701128402549162, −12.22632932769452309415921776111, −11.08072694554883594543953958216, −9.251772154646168503253955082345, −8.338114631365679763179574710382, −7.38409377127990473113029651320, −6.60041205018225633938741962350, −4.53563908096151620515128822542, −2.08924396664372077928137107283, −0.50234956494031979924896954793, 2.73565817666765947748448248419, 3.77772681899019996222911501840, 5.33705633997460280100190425972, 7.69798363971584084999831637139, 8.540053846707439987639115073271, 9.937654927788981200168361841670, 10.38560892502360040294394818248, 11.56791737085733597757842921526, 12.76534260694946002413753111178, 14.49791841409621753305684604617

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