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Properties

Label 2-82-41.38-c4-0-0
Degree $2$
Conductor $82$
Sign $-0.933 + 0.358i$
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 + (−9.30 − 3.85i)3-s − 8i·4-s + (16.4 + 16.4i)5-s + (26.3 − 10.8i)6-s + (32.0 + 13.2i)7-s + (16 + 16i)8-s + (14.3 + 14.3i)9-s − 65.8·10-s + (−5.87 + 14.1i)11-s + (−30.8 + 74.4i)12-s + (−235. − 97.4i)13-s + (−90.6 + 37.5i)14-s + (−89.7 − 216. i)15-s − 64·16-s + (−147. + 61.2i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−1.03 − 0.428i)3-s − 0.5i·4-s + (0.658 + 0.658i)5-s + (0.730 − 0.302i)6-s + (0.654 + 0.271i)7-s + (0.250 + 0.250i)8-s + (0.177 + 0.177i)9-s − 0.658·10-s + (−0.0485 + 0.117i)11-s + (−0.214 + 0.516i)12-s + (−1.39 − 0.576i)13-s + (−0.462 + 0.191i)14-s + (−0.398 − 0.962i)15-s − 0.250·16-s + (−0.511 + 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $-0.933 + 0.358i$
Analytic conductor: \(8.47633\)
Root analytic conductor: \(2.91141\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :2),\ -0.933 + 0.358i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00597952 - 0.0322574i\)
\(L(\frac12)\) \(\approx\) \(0.00597952 - 0.0322574i\)
\(L(3)\) 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 \( 1 + (1.24e3 - 1.13e3i)T \)
good3 \( 1 + (9.30 + 3.85i)T + (57.2 + 57.2i)T^{2} \)
5 \( 1 + (-16.4 - 16.4i)T + 625iT^{2} \)
7 \( 1 + (-32.0 - 13.2i)T + (1.69e3 + 1.69e3i)T^{2} \)
11 \( 1 + (5.87 - 14.1i)T + (-1.03e4 - 1.03e4i)T^{2} \)
13 \( 1 + (235. + 97.4i)T + (2.01e4 + 2.01e4i)T^{2} \)
17 \( 1 + (147. - 61.2i)T + (5.90e4 - 5.90e4i)T^{2} \)
19 \( 1 + (411. - 170. i)T + (9.21e4 - 9.21e4i)T^{2} \)
23 \( 1 + 365. iT - 2.79e5T^{2} \)
29 \( 1 + (1.10e3 + 457. i)T + (5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 - 403. iT - 9.23e5T^{2} \)
37 \( 1 + 812.T + 1.87e6T^{2} \)
43 \( 1 + (1.11e3 - 1.11e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-778. + 322. i)T + (3.45e6 - 3.45e6i)T^{2} \)
53 \( 1 + (-1.24e3 + 3.01e3i)T + (-5.57e6 - 5.57e6i)T^{2} \)
59 \( 1 - 3.92e3T + 1.21e7T^{2} \)
61 \( 1 + (1.97e3 - 1.97e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (505. - 209. i)T + (1.42e7 - 1.42e7i)T^{2} \)
71 \( 1 + (2.26e3 + 937. i)T + (1.79e7 + 1.79e7i)T^{2} \)
73 \( 1 + (-3.59e3 + 3.59e3i)T - 2.83e7iT^{2} \)
79 \( 1 + (1.51e3 - 3.65e3i)T + (-2.75e7 - 2.75e7i)T^{2} \)
83 \( 1 + 2.91e3T + 4.74e7T^{2} \)
89 \( 1 + (8.96e3 + 3.71e3i)T + (4.43e7 + 4.43e7i)T^{2} \)
97 \( 1 + (-5.37e3 - 1.29e4i)T + (-6.25e7 + 6.25e7i)T^{2} \)
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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

−14.58135682475857765381602097075, −13.01529672029465076613667552067, −11.90472447300036418528218141948, −10.80252156459891209675241665713, −9.950989458398724836547233318507, −8.397943467335411391607147372272, −7.03755549246459092371654883278, −6.11973233505212265374534839930, −5.02983610367471202697987304371, −2.09870093189553539353228178990, 0.02036330294038793508328596142, 1.94518867844849218722168930273, 4.49798339250215341143956244534, 5.45592794241889922176820925571, 7.16239135306709171965635562856, 8.758377659130957692446154315829, 9.783316722483932212176455404559, 10.85580996514238128272805467063, 11.63205296927901947140679419274, 12.71278704894908028820100315476

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